Before I moved to NYC and I used to come in just for a few days to see shows and such, I stayed at the Vanderbilt YMCA. It is an excellent place to stay. The rooms are tiny, so keep that in mind, but they're very clean, air-conditioned, and there's a cable TV in every room. When I used to stay there, like 3 or 4 years ago, I paid $70/night. I don't know what they're charging nowadays, but you should book your reservations online as opposed to calling them, because the prices online are a bit cheaper.
http://www.ymcanyc.org/ There's also one on West 63rd Street, which is about the same distance to Broadway. But either way, you can easily walk to the theater district from either. From the Vanderbilt it's a few avenues, but not a big deal. From the West Side one, it's obviously about 13 street blocks to get to the Gershwin.
"Winning a Tony this year is like winning Best Attendance in third grade: no one will care but the winner and their mom."
-Kad
"I have also met him in person, and I find him to be quite funny actually. Arrogant and often misinformed, but still funny."
-bjh2114 (on Michael Riedel)