The Eliza Jane is a cool hotel just outside of all the bustle of the quarter but in Easy walking distance. If you want something other than the quarter, Hotel Pontchartrain in the Lower Garden District is cool and you can take the streetcar up and down St. Charles to get to the quarter and see all the cool houses in the garden district. There's also a cool hotel attached to the WWII Museum, it's some kind of Hilton but I don't remember its name.
As far as safety goes, aside from the normal shenanigans in the Quarter, anything south of Esplanade and North of highway 90 between IH 10 and the river is decent. I personally wouldn't rent a house any further north than Esplanade but I don't remember the immediate area being too bad. Anything along Canal St. is going to be busy and expensive. Depends on what vibe you're going for. The Roosevelt is very cool and has a lot of history.
Along St. Charles and Magazine streets are mostly okay, the garden district in general is decent. I'd stay away from anything north of IH-10 towards the lake, and for the love of God don't book anything anywhere close to New Orleans East (I.e. over by the Lakefront Airport).