I'm not a student there, but I know that some students live in University Towers and New Haven Towers. They're about a 5 minute walk from Yale Med's "campus" and hospital.
The towers next to the medical school are pretty pricy but it all depends on each one's financial aid I guess, and whether folks are pairing up with roommates.
A lot of students choose to live on campus during the first year, but the towers (Crown, Crown Court, and Madison) are also a common option. They're fairly expensive with studio apartments alone ranging from $1125-$1395:
http://newhaventowers.com/rate-sheet/. If you share an apartment, the cost is split. In my experience, living in downtown New Haven is going to be around $1000 if you live in a studio, with 1 bedrooms being somewhat more expensive. If you have roommates you might pay anywhere from 500-1200 for a room, depending on the building -- closer to 1000 is more common. Buildings that are cheaper (the Elmhurst, anything owned by Pike) tend to have pretty terrible management, which can be a problem when your heater breaks in the middle of winter, a frozen pipe bursts, or you have a ceiling leak. Pricier places tend to have good, reliable management (like Elm Campus and the Towers) but you might pay up to 1,200 for a single bedroom in a shared apartment. There are some older houses where you can lease a room along Howe, George, and Dwight. These tend to start around $500, but that really depends on what kind of privacy and set-up you're looking for. If you're down to live with 3-4 other people, I'd check craigslist for those options. Unfortunately (I hate to be the bearer of bad news), New Haven rent prices have consistently surprised me... and not in a good way.
Also! If anyone has any questions about areas of New Haven -- which ones are safe etc. -- please feel free to PM me with apartment listings/addresses etc. I have a good amount of experience (multiple years) with the city and have lived in multiple locations with multiple landlords.
(Edit: just to clarify, I think New Haven is overall a pretty safe city, though I am from a very urban area. Some areas will have weird random occurrences like the shooting that happened on campus over the weekend, but there are other areas that should be avoided.)