O ACLU notes que:
The Transportation Security Administration (TSA) acknowledges that photography is permitted in and around airline security checkpoints as long as you're not interfering with the screening process. The TSA does ask that its security monitors not be photographed, though it is not clear whether they have any legal basis for such a restriction when the monitors are plainly viewable by the traveling public.
The TSA also warns that local or airport regulations may impose restrictions that the TSA does not. It is difficult to determine if any localities or airport authorities actually have such rules. If you are told you cannot take photographs in an airport you should ask what the legal authority for that rule is.
Popular among photographers, The Photographer's Right, a summary of legal rights related to photography (prepared by a lawyer), says that:
There are some exceptions to the general rule. A significant one is that commanders of military installations can prohibit photographs of specific areas when they deem it necessary to protect national security.
Despite misconceptions to the contrary, the following subjects can almost always be photographed lawfully from public places:
accident and fire scenes, children, celebrities, bridges and other infrastructure, residential and commercial buildings, industrial facilities and public utilities, transportation facilities (e.g., airports), criminal activities, law enforcement officers
Some airports are more spotter-friendly than others. You can search online to find discussion forums on people's opinions which ones are better than others. Any airport that is private property (not many major ones, I believe) of course can do whatever they want, but many airports stop photography for a "safety" measure.
Standard IANAL disclosure.