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Don't do either 143rd or 173rd, and don't do a hostel, if you can help it.
Luckily you're female, and I'm guessing that you're fairly young. Do some Googling for the lists that exist for inexpensive places to stay in Manhattan. There are some boarding-house types of places, including a rather shabby one at a terrific location in the East 30s. Or, as was said, see about renting a room, perhaps with a student; you can check Craigslist (but be cautious about scams) or even check with particular schools. As a matter of fact, ask your own place, wherever you're taking the class. And there are one or two forums specifically for ex-pats in NYC, where there might be some suggestions.
i'm a brit myself. I had a sublet through craigslist that went very well. Otherwise I stayed at Spot hostel in harlem about 127 and lennox - very clean, tv, computers friendly, lockers etc