I was at the medical school last year for grad school and most people I knew lived in the Garden District, which is the part of town where the undergraduate campus and law school is. Most apartments in that part of town are great!
The medical school is downtown, so apartments there will be much priceier, but safe.
I lived in the Marigny area and I loved it. That neighborhood is right outside of the French Quarter but still pretty close to downtown.
If you have any specific questions feel free to PM me!