Bear Mountain Bridge Motel, Fort Montgomery, New York, about 2 miles north of the Trail on US-9W just before you cross the Hudson. Not a hostel, but a cute little motel with 6 inexpensive and clean rooms.
Interesting story about coincidence -- I'm trying to talk my younger brother into joining my 65-year old mom and me for 4 day hikes covering the 45 miles in central New York that I haven't walked. He agrees on the condition that we stay someplace inexpensive. I call around and finally find the Bear Mountain Bridge Motel for something like $45/night.
My brother works as a real estate underwriter for an insurance company south of Syracuse. The afternoon before he takes off to join us, he gets a request to insure a property located in Fort Montgomery. Not trusting the agent, he asks for photographs of the property and will decide when he returns.
My mom lasts for 4-6 miles each day, when she gets off the Trail to rest (and shop one day!) while my brother and I hike on for another 8-10 miles.
Upon my brother's return to work, he receives a packet in the mail of photos of the motel property he's being asked to insure. He recognizes the motel as the one where he just spent 3 nights! Looking a little closer, he comments to himself that the car parked out front looks like his car! Looking even closer, he recognizes our mom sitting on the front porch talking to the propietor! The pictures had been taken one afternoon when our mom returned to the room after hiking. Talk about coincidence!