I've been in NY for the past 6 years doing undergrad, working, etc. Never with any real amount of money. It's worked out fine for me and a lot of other people I know.
1. Apartments: Cheaper apartment shares that aren't in Manhattan. Astoria in Queens, and several areas in Brooklyn are nice and very quick on the subway (10-20 minutes) often have shares (your own room in a 3-4 bedroom apartment) for $500-600. If you want to spend more, there are other options ... but you can get by on that much.
2. Transit: You don't need a car so the main transportation expense is only $76 for an unlimited monthly metrocard.
3. Food: Sure, there are lots of expensive restaurants, but there are also a lot of places to eat cheaply. There are often really good / cheap lunch specials, and there are always some inexpensive options in any area. Cooking helps a lot too. Watch out for super overpriced Manhattan supermarkets, but freshdirect.com is a good alternative with cheaper (but good quality) groceries and delivery.
4. Entertainment: Again, plenty of expensive options, but also a lot of free / cheap ones. You can take advantage of student rush tickets at the opera/ballet/symphony and other places. Wandering around and hanging out at Central park are both free. There are tons of other things to do like that, too ...
It's not easy, but very doable. I guess at the end of the day New York can be really expensive if you want it to, but it doesn't have to be.
