With 80% of the city flooded, there were very few areas NOT affected by Katrina. However, areas along the "crescent" of the river...Most areas south of historic St. Charles remained dry. Uptown, Garden District, French Quarter would be the 3 I suggest. Uptown is a quick 5 minute drive down Claiborne (when they get the traffic lights working again) to the medical school. Garden district and French Quarter are closer.
Rent is going to be higher than normal, due to the demand on the housing market. I haven't started looking for another place down here, so I don't know how terrible it really may be....but I know finding a new place in mid-december was a pain in the ass. Stay along the river, its the only semi-safe place in the city. (and I don't mean crime, I mean water). Personally, I've enjoyed living uptown, and the garden district isn't that much different...they kind of blend together, except the garden district seems to have (had) a higher crime rate.