For NYU students, I'd suggest looking for housing close to trains that you can take to campus. I have a friend from NYU who loves the N train so she looked in Sunset Park solely because it was serviced by the N. The A/C/E and B/D/F/V will also be good options for you. Living off an express stop is better than a local stop in my opinion. So that gives you the A and the B/D along with the N. Neighbourhood wise in Brooklyn, you have the option of Park Slope, Ft. Greene, Prospect Heights, and Sunset Park. Each of these neighbourhoods will be a 15-20 minute train commute to West 4th/8th Street stops.
For the person who inquired about living in a studio, I'm not sure how much you're expecting/able to spend on rent so it's hard to say but studios are expensive. In the neighbourhoods I mentioned, I doubt you'll find a studio under $1,000. I definitely haven't seen it if it exists. If you search studio in Brooklyn on Craigslist, it'll give you a range of prices. That would be a good place to start thinking about what you can afford. Then, you can figure out whether the neighbourhoods where those studios are will be desirable to you.