It'd likely have to go largely underground. This is the approach being taken for Dublin Airport (again, a far, far smaller city than NYC); it'll be served by a largely underground metro line, running every 3 minutes each direction, taking about 20 minutes from the city centre.

Now, the catch there is that this metro isn't going to the airport, it's going _through_ the airport. Even without the airport it would be justifiable, so the airport kind of gets it for free. That's probably the only scenario where you can justify this sort of thing; it would be comically overkill if it was just to serve the airport (it will be able to move 20,000 people per hour per direction, which is... a lot more than the airport can move.)

That said, you'd think something along these lines might be justifiable; as you say, the area surrounding JFK is dense.