I've gone the last two years with friends from college. We've always stayed in Cocoa Beach and probably will again this year. If you stay on the beach you can count on a 30-45 minute drive to the stadium depending on traffic. We stayed at the Best Western last year and the La Quinta Inn the year before that. They're exactly what you'd expect...a room with beds but not much else. I can't say whether that area is considered safe or not, but we've never had any issues.
If you stay beach-side, stay south of Patrick AFB, Cocoa Beach gets a lot of tourist traffic....oh, wait, you'll be a tourist.
I'm sure we've had some of these discussions before and gotten a lot of good advice out of then, in general, I highly recommend getting down for Spring Training. The beaches are great, good surf, plenty of nice places to eat. Viera is like every large new development you've ever been to, all the chain resturants you can think of from McDonalds up through Longhorn Steakhouse to World of Beers.
Access to the players is amazing. Just bring plenty of sunscreen and remember that Florida can get a bit chilly in February and March
