it depends. some people, esp students live downtown right near the campus and do not have cars. that being said, a lot of residents choose to live in the graduate student neighborhood (east rock) which is a great area of old victorian houses converted to apts with tree lined streets and a big park. Others, especially ppl looking to purchase condos or homes live in the suburbs around new haven like hamden, and branford ct.
if you live downtown, you dont *need* a car, and there is a shuttle to the hospital from the east rock, but i found a car helpful to get groceries, go to the mall or boston post road, get out of town on the weekends, etc... if you live in east rock or hamden you definitely need a car to get to work on time. parking at the hospital is subsidized for residents and car insurance isnt too bad in CT (but im from nyc, so not a great judge).
good luck,
wex