Since Mt Sinai is on the Upper East, Columbia the Upper West, Cornell the mid-Upper East, and NYU the Midtown East, I would stay close to *one* of the schools and then train around to the rest. They are all readily accessible via the subway, so my advice would be minimize your distance to one and just switch lines for the rest. Be sure to call in advance and find out what the most effective method of line-switching is for each school (I believe to get to Mt Sinai, I took the Green line #6 which lets you off about 2-3 blocks from the school; I think Columbia is one of the last stops on the A train which goes through Penn Station; I walked from 33rd and 5th to NYU; and I've never been to Cornell).
If you're going to be staying with students hosts at 'some,' then I guess you don't need to worry about locations for those schools and since we're only talking about four schools, and assuming that 'some' implies at least 2, then theoretically you'll only be travelling to one school if you follow my advice.
Buena suerte.