We have flown in to Stewart and Newark. My advise: Fly in to Stewart and rent a car! SO much easier. Smaller airport - less congestion. About a 20 minute drive to WP on roads that aren't congested and quite a beautiful drive over Storm King Mountain. It is always nice to have a car even if you stay at The Thayer. We have driven over to Central Valley which has a nice upscale outlet mall. Also you can drive to surrounding towns (like Fishkill, Hyde Park and Cold Springs) which have many more options for restaurants than Highland Falls. Flying in and out of Newark was a hassle for us. We aren't used to big city driving and traffic so it was not easy for us. Several toll booths that you have to go through. Comparing the drive from Stewart to WP and the drive from any NYC airport to WP, hands down we would choose Stewart. For some, I've heard that Stewart is a lot more expensive for them than the NYC airports but for us it is about the same or not significantly more. Another small airport option is White Plains. A little further than Stewart but not as far as NYC. I've never flown in to there so I can't say what it is like.
If you want transportation from NYC to WP, there is a bus line that leaves the Port Authority Bus Terminal in the Theater District. It takes you to the Visitors Center-which is close to the Thayer Gate. You can also take a train from Grand Central Terminal to Garrison (across the river from West Point), but you will need to prearrange a cab to pick you up there to take you to your hotel.
After many trips to WP over the years, here is our preferred iterinary.
Fly in to Stewart, rent a car, stay at the Homewood Suites there at the airport or The Sierra at Fishkill or at The Thayer. The Sierra is very nice by the way and very reasonable and includes a great breakfast. Between our visiting times at West Point, we like to shop at the outlet mall, visit the cute little towns around West Point and a couple of times we have taken the train from Beacon ( near Fishkill) to NYC and to see a play, shop, eat and then take the train back. It is about a hour train ride and beautful as it runs along the Hudson. Doing all that makes it necessary to have a car.
This is MUCH more information than any of you have asked for, but I got on a roll! Sorry! The Hudson River Valley is quite a beautiful area - so take advantage of it while you are there!