I am graduating from SUNY Upstate this May. I have lived here for 4 years, so here is the lowdown. Stay away from the dorms - it's called Clark Tower and is about a 5 minute walk to the hospital, but they are horendous. No air conditioning, old rooms with old carpeting, light fixtures, relatively pricey etc. Freshmen tend to live there because they don't know any better when they start, but almost 100% of them move out the subsequent year.
There are 2 towers within about a 10 minute walk to the hospital. They are called Madison and Jefferson Towers, and they are very nice. Jefferson is the nicer of the two, but it is slightly more expensive. Madison is nicer than Clark, but they have a few problems: central AC, so they turn it on only when it's really needed, sometimes the hot water in the shower is unpredictable, and there are some weird people living in the building. However, they do have a parking garage, which Jefferson does not. Madison also has townhouses, which are sweet. In Jefferson, every room has separate AC and the rooms and lobby are a lot newer and nicer. Madison has 1 bedroom, 2 bedroom, and townhouses. Jefferson has studios, 1 bedroom, and 2 bedrooms. All the apartments have plenty of room.
Both of these buildings are within walking distance of downtown, Syracuse University, and the main hospitals. They have about 80% medical students and residents living there, but few SU students, which is good. I'm not going to say they are in the safest part of Syracuse, since there are numerous government-funded assisted living apartments nearby, but you can't have everything. Both buildings tend to fill up very quickly and rent can fluctuate, so call as soon as you can.