Really depends where they live. And NO. You will NOT save money by living close to school. If money is a priority that is a big no no. Commuting by subway in New York City costs about $100 a month. I'd say 80% of people commute by subway. What you want to do is live in a neighborhood that is reasonably away from Manhattan but at the same time cheap.
Example: One friend I know lives on 52nd st and 3rd ave. She pays $3800 for a STUDIO. Sure you can walk to school and be there in 15 minutes. (And yes, you can find obscure studios for like $2000 a month with no windows that another one of my friends lives in.)
Another friend I know lives in Jamaica, Queens. Lives with a classmate and they share a 3 bedroom HOUSE. He pays $600 a month. HUGE difference. So that is $600 + $100 for commuting although the commute is a little over an hour.
Personally, I'd find something in the middle like 30 minutes away and like $800-$1000 to room with someone. That is why Astoria, Queens is so popular with my class.