The crime in the Bronx wasn't as bad as it was back in the 80s and early 90s (thank the crack epidemic for that). Gentrification is at full blast there now. However, crime still remains, although a shadow of its former rate.
Some parts of the Bronx, over to the south aren't good for first timers or tourists, but they are somewhat passable. Most of the crime in the Bronx, and New York in general, is drug related, and if you aren't interested in taking a whiff of that stuff, then you might not get bothered.
Neighborhoods like Throgs Neck, Riverdale, Kingsbridge, Country Club, and Woodlawn are amazing, yet quiet places.
Neighborhoods like Hunts Point, Tremont, Morrisania, Mott Haven, and Melrose are a little rough around the edges.
Former New Yorker :(