My personal JFK
(1) Terminal 1, upstairs at the Food Court. You do NOT need a boarding pass to enter this area. The food choices are sketchy (although the Chinese food isn't bad, just expensive) and you can grab a window seat and have a decent view of 31L/13R. This also includes a decent view of the T1 ramp. The runway view is not perfect - there are minor obstructions in the way - but basically it is a decent place to watch planes taking off. I am not sure if it is great for photography.
(2) AirTrain station at Terminal 6. At the far end by the elevators, near to T6
, you are closest to runway 13L/31R but you have a great view of T6
, as well as the Saarinen building and some of T4
. Granted most of the local traffic is all JetBlue, but there are decent views afforded when 31R is being used for landings. Often aircraft will taxi by going to T7
) and T9
Otherwise, there is not a lot on-airport. My experience is that you will get funny looks from the PANYNJ police if you loiter too long at the airport terminal sign or anywhere in that area.