I stay at the Y on 63rd Street, the one that the Village People song is about. It's $85 a night and you share a bathroom. The room is essentially a cell, but who cares? It's a half block off Broadway and Central Park West, walking distance to Lincoln Center, and about 10 blocks from Times Square. Plus, you get all the Y facilities you need: pool, sauna, etc.
You can book online, the rooms are clean and it's a nice place to stay.
NYC Visitor and Broadway Fan