I finished an MA at the UW last june, and lived in the area between Wallingford and Freemont. It's on several bus lines that go straight to campus at pretty regular intervals, at least during the day (quite a bit of the seattle public transit system shuts down after 7pm or so. You can still get most places, but you'll end up transferring or waiting around more), and it was maybe a 30-40 minute walk or a 10-15 minute bikeride. If you have a bike, I'd really recommend finding a place along the Burke-Gilman trail -- it was one of my favorite things about living in Seattle, and a really nice commute to campus.
Greenlake, Ballard/Phinney Ridge, and Ravenna are all along good bus routes. Parts of Queen Anne, Capital Hill and the Central District are also really reasonable without a car. I never drove, and never had too much trouble getting anywhere I needed to go.
You might also think about what kind of neighborhood you want. Freemont, Wallingford, Ballard, Greenlake and Ravenna are all a bit quieter; Queen Anne, Capital Hill, Central District and the International District tend to be more 'urban'.
