I interned in midtown a couple years ago. Most of the colleges there (NYU, Colombia, etc.) offer summer housing for interns, but they are probably full by now. Sounds sketchy, but check craigslist. That's how I found mine. Getting a place by yourself is extremely expensive, especially in midtown (2500-3000 per month minimum), but if you find someone on craigslist that is subletting a room, you can probably pay about half that. Also check nyhabitat.com.