Definitely try to get set up for a Columbia apartment before you get to NYC (Columbia owns a ton of properties). It definitely helps, but it might delay you finishing your degree.
Affordable apartments are hard to come by, and you may not want to leave it.
I would try to get a place in Morningside Heights or UWS or Harlem. I would not recommend living in Brooklyn, unless you enjoy fricken' long train rides.
I hate that, especially if you want to do something at night at the University. After finishing at Columbia, I lived in Washington Heights for a while, but it definitely felt more residential than lively, and it was terrible when you wanted to do things with friends downtown or in Brooklyn. UWS, Morningside Heights, or Harlem would be most convenient.
Best of luck. I enjoyed my time there, though I was only at Columbia for a master's degree.
I love NYC and lived there for about ten years. Sometimes it was difficult (finding places to live, one year I moved 7 times), but other times it was really quite wonderful. I miss the aliveness of the place.
Oh, you really missed out there. They have so many properties, though. Possibly it could still work out.