Certain characters can perform a bare-handed parry

The exact technique of catching a sword between one's bare palms in this manner is não explicitly listed in the game rules. However, some characters possess the ability to deflect or block a blade bare-handed.

• The Battle Master fighter can learn the maneuver Parry, which allows you to reduce the damage of an incoming hit, possibly to zero damage. You do not need to be armed to Parry.
• A monk of third level or higher can catch and deflect incoming ranged weapon attacks, such as someone throwing a blade, but not a melee attack from something like a sword, which is usually what is described in a sword catch situation.

Em termos de coisas qualquer character might try, which may represent a sword catch:

• The Dodge action (PHB p.192) allows the player to focus entirely on avoiding attacks, spending one's entire action, and the result is that all attacks made against you have disadvantage (and you make Dexterity saving throws with advantage).
• Ready an action to grapple, although this doesn't represent a block, but rather the after-effect of grabbing the enemy by the sword

Some monsters have a Parry action which allows them to increase their AC against one melee attack which would hit, as a reaction. However, this is a monster-specific ability, and also require them to be holding a melee weapon.

I'd recommend using something like the dodge action. You have to spend your entire action, but if you do, the opponent has disadvantage. If that causes him to miss, good job! You caught the sword.

This makes more sense with a character like a monk. However, a non-bare-handed catch or block wearing gloves or gauntlets, or even grabbing a non-moving sword bare-handed, is surprisingly feasible. See the Youtube videos Historical Perspective on Blade Grabs - Showcasing HEMA e Half-swording - Why grabbing a sharp blade in a sword fight is not crazy.

RAW, this cannot be done

There are no rules that I am aware of that allow such a thing to be done.

A monk has the closest related feature

A level 3 monk has the class feature Deflect Missile, which says (PHB, p. 78):

Starting at 3rd level, you can use your reaction to deflect or catch the missile when you are hit by a ranged weapon attack. When you do so, the damage you take from the attack is reduced by 1d10 + your Dexterity modifier + your monk level.

If you reduce the damage to 0, you can catch the missile if it is small enough for you to hold in one hand and you have at least one hand free. [...]

However, this only works with ranged weapon attacks, so this would not stop a sword in the way that you describe.

Just narrate blocking this way

If this is just for flavour and the player does not expect to actually disarm the opponent or anything, you could simply describe a near miss being stopped by the PC in this way. That way, mechanically, the enemy simple missed, but the player gets a cool narration.

As the DM, you are responsible for setting and enforcing the tone of the campaign

As NathanS e Assistente quadrático have noted, a character might say "I catch the sword in my hands" as a way of narrating a dodge or parry. This is mechanically sound within the rules as written, but that does not mean that you as the DM are obliged to allow it.

The DM's Authority to Set the Tone

As the DM, you are largely responsible for setting the tone of the campaign world. For example, you can decide if the world is high or low fantasy. Often, this decision is made in collaboration with the players, to make sure the setting will be something they are interested in. In recognition of the higher time/effort investment required in the DM role, however, the DM is sometimes allowed a certain amount of supremacy in this: after all, you can't force the DM to create a world that they don't want to create, and if a player feels strongly about playing in a certain setting, they should volunteer to create it themselves. All of this comes down to the social norms of your particular group.

Once a tone is agreed upon, it is your job to enforce that tone. If your group agrees on a gritty realistic Euro-medieval setting, and a player narratively describes dodging an attack with an anime-style wall run, it's up to you to shut that down. If your group agrees on a fun, goofy, gonzo adventure-romp, and a player narrates a melee attack in which they castrate an enemy, it's up to you to shut that down. The DM must maintain the integrity and immersiveness of their world; part of that means making sure the players exist within that world's reality.

Sua situação

Ideally, there is a discussion about setting tone and expectations during Session Zero. If there was a decision at the start of the campaign that this world would not be the type of place where people can catch swords, remind the player of this. If your group did not have a Session Zero, or have otherwise never discussed expectations of tone, then now is the time to decide. Consider the previous section in this answer: what is the norm for DM power in your group? Is this a decision you as the DM would typically make on your own?

From your post, it sounds like you might be open to giving the players some discretion in setting the tone. If this is the case, consult the entire group to see how fantastical they want the setting to be. Once a tone is agreed upon, you will be better equipped for adjudicating your players' narrative descriptions.

"How I should respond to..." depends on what kind of game you want to run

Every DM has their own style. Some like their games to be realistic, some like to strictly follow Rules as Written, some are avid believers in the Rule of Cool, and some don't really care about the rules at all. But whichever of these styles you play, that is provavelmente the style that você enjoy, and since the point of the game is for everyone, DM included, to have fun, then you should (probably) respond according to whatever style it is you play the game.

Now, "play according to your style" sounds nice, but that doesn't actually tell you much, does it? So, I'll use your example to try to illustrate a few different styles.

Let's start with RAW

As Assistente quadrático e NathanS pointed out in their answers, there aren't actually any rules that (directly) support this action. There simply are no RAW mechanics for catching a sword like this. But, as Quadratic Wizard also pointed out, there are quite a few rules you could easily use as a stand-in, and then follow NathanS's advice and just narrate the action of actually catching the blade with your hands.

Some rules that would allow you to do this are (again, pulled from Quadratic Wizard's answer) the Blademaster's Parry, the action-consuming esquivar, or readying a Grapple. However, as both Quadratic Wizard and NathanS have noted, you still have to narrar this actions as if the sword were caught.

Rule of Cool

The Rule of Cool can be one of the most entertaining parts of a game. The rule is, if it's cool, then let it happen. It's not an official rule in qualquer sense, but I know quite a few DMs who nevertheless use this "rule" as a core guiding influence in their games. It makes it easy for players to shine and have their moments in the spotlight, and it's a pretty good way to make players feel like their characters really are awesome adventurers above and beyond the average commoner. Following this, a low dexterity (acrobatics) check of DC ten or fifteen might be appropriate for attempting to catch a sword.

Isso não geralmente work well for very serious, realistic, or gritty games though, because it tends to break the tension and immersion. So if ridiculous and over-the-top awesomeness doesn't fit the style you're running, then this is probably not the solution to go with.

Here's how I would rule it

My own style is that I try to follow the rules as much as I can (in part because I am still trying to learn them), but I am willing to bend those rules whenever and wherever I think that I can make it more fun. So with that in mind, I would allow the attempt to be made.

That said, it is almost impossible to catch a blade swung at you. Your strength, reflexes, and luck have to be phenomenal to even have a chance! So with that in mind, I would set the DC to 30. 30 is right up there on the edge of what a high level PC poder be able to accomplish, if the appropriate skill is strong enough and they get lucky, so that seems like an appropriate difficulty to me. Additionally, since this is a reaction to an incoming attack, this will eat up the character's reaction for that round.

Now, to actually catch the blade, an individual has to be forte suficiente para completely stop a blade mid-swing using only the friction between their hands and the blade! That individual also has to be rápido enough to catch the blade mid-swing, with their hands closing on the blade. Close your hands too soon, and they get sliced! Close them too late, and... well, you get a sword straight to the face.

I don't like making two checks to perform one action, even though both força e destreza apply here, so I won't do that. Instead, I would tell the player to make ou a Strength(Athletics) check or a Dexterity(Acrobatics) check to catch the blade. This does benefit the player a little bit, especially since I'm calling for a DC 30 check.

Once the check has been made, there will be consequences. I want those consequences to be interesting on both failure and success, so the player feels like their choice actually did something, so I'm not going to leave this is just a narrative fluff.

So, on a success, I would rule that the player actually manages to catch the blade mid-swing! The shock and surprise on the attacker is so great that all sources gain advantage against that target until its next turn! This grants a clear and powerful gain from a very difficult maneuver, rewarding the player if they can actually pull it off!

On the other hand, failing to catch the blade means that you straight up take a sword to the face, with no attempt to dodge, so... if you fail to actually catch the blade, then the hit is an automatic crit. This also means that an attack that your AC teria ignoradas torna-se a hit because you tried to catch the blade. This is a bit harsh, I'll admit, but the character is literally just taking a sword to the face without all of the abstracted blocking and dodging that normally goes into landing a hit.

Of course, the most important part of this would be for me to tell the player a.) that this will be an incredibly difficult check, and b.) exactly what the consequences are for success or failure. This way, I'm not blindsiding them with a crit out of nowhere, and the player can make this decision with full knowledge of both risk and reward - an essential part of making an informed decision.

Now... a DC 30 is very high, and would exclude all but the strongest or most dexterous of characters. If I decide that I want this to be an option more often, then I think I'd make a dynamic difficulty check. An easy way to do this might be a difficulty check of the enemy's modified attack roll + 10 (so, a roll of 12 with a +5 to hit mod (that's a 17 versus AC) becomes a DC 27 Athletics or Acrobatics check to catch the blade).

Strict realism

Of course, if you're going for strict realism, then you have the easiest answer of "No, it's impossible." While this might be a little bit less fun for the player, it is a reasonable, and easily defended, ruling that doesn't require either further justification or any remotely taxing decision-making. DM-ing can be very taxing, and every decision drains your creative well just a little bit. The more complex the decision is, the more draining it can be, so it is perfectly reasonable to look at a (near) impossible task and just say "No, your character is not able to do that."

There is no particular way you deveria responder

The TLDR of all this is that everyone has their own style, and everyone thinks different things are fun. Choose an approach that fits your game and your style, make it fair, and stick with it!

My usual response to this sort of question comes in four parts

• In DnD, you don't usually have specific blocking moves. I'm usually a fan of "if you want to try something, the GM finds a way to let you try, even though success may not be guaranteed." But you're ALREADY trying to stop opponents hitting you, so it's unfair if you can do that even more just by asking the GM for it, without either being at the expense of your attack (as with the dodge action, and not strong enough it's usually preferable to regular combat), or a particular ability (you can't just let everyone catch arrows, or the monk's ability is useless). Try to encourage players to usually use active combat actions like attacks, not defences, because that's how DnD usually works.
• You also need the GM and the players to have a shared imagination of what's possible, so people don't say "but that's impossible" all the time. That involves some amount of "there's a particular rule for that, you can't just do it". But also, if the players have cool ideas, it's good to include it whenever you can, it makes a richer world than only the things you can invent.
• I'm a big fan of including cool things. If this is a really dramatic moment, like the character is unarmed and is being executed by the Big Bad Evil Gal/Guy, give them a one off roll or any ability that seems appropriate, to see if they can pull this out of nowhere.
• If the players are excited to include that particular thing, you could invent something, but if it's negating damage from an attack, you need to make sure it's not something you can do all the time. Possibilities include:
  • Spend an action to do have a reasonable chance but not certainty of doing this, like the dodge action or a better version of it. That makes sense being something you only do when you're desperate, but will probably feel annoying that the players need to stop attacking to do it, which is usually a mistake.
  • A custom ability or feat for one character who wants to take it. Say, only once per long rest you can negate an incoming melee attack, and maybe some other bonus like the enemy has to make a strength check to make another attack.
  • Design a house rule for everyone, or an ability, or just a convention that you describe things a certain way, like "when an attacker rolls a 1, you can make a dex check to grab their sword" (either purely as flavour, or giving some other benefit like preventing them making other attacks for a turn) or "if an attack would reduce you to 0hp, you can make a dex check to catch their sword, if you succeed, go to 1hp instead"