Hi! I got my B.A. from FSU and lived off of Jackson Bluff my senior year. I think if you live off of that street within about a mile of the school you will be fine. I know it may look sketchy (that was my first impression too as I had always lived in the dorm or a big, corporate run apartment), but it is really not bad at all. The houses are perfect for graduate student living (something I miss in SoFL) and the further you get from the actual campus and stadium, the quieter it is. The neighborhoods are great because you get a mix of tons of college students and Tallahassee residents, so you know there are people around year-round even when you've left for vacation. And, you are close to virtually everything except the mall. I would, however, stay away from Frenchtown as it is not that close to the school and not the best area of town.
Also, even though Blairstone is further away, there are some really nice apartment complexes off this road. One that comes to mind is Arbor View. I think they cater more to young professionals, so if you do have a car, don't want to be around crazy undergrads, especially on game days, this might be something to look into.