Its closed now!......But I think most would agree that Hong Kong's Kai Tak was probably the most photogenic airport in the world. It was not so much the airport itself that was stunning, but Kai Tak's locaton, the dramatic mountain and cityscape backdrops the endless parade of widebodies. These elements drew photographers from the world over, just to shoot at this airport.
One airport here in the US that is interesting to shoot would be BOS, just check out some of Mark Garfinkel or Vasco Garcia's work. With its close proximeity to downtown BOS, the skyline provides a dramatic backdrop. As well as a very diverse selection of international and domestic service.
From an architectural stand point I would say that DEN, CLK are some of the more interesting airports, as well as Daharan, Saudi Arabia.
Hope this helps, I am sure that there will be additional airports added by other photogs.
"Show me the Braniffs"