I am not a big fan of Nassau, but I have never felt unsafe. Just follow the normal precautions that you would in any city (keep your valuables secure so you cannot be pickpocketed, stay in populated areas, etc.). The biggest annoyance is aggressive vendors (as Cruise Director Jacqui said, the hair braiders will remind you of all those seagulls in "Finding Nemo" yelling "Mine, mine, mine!"). They might also try some minor scams, like kids singing for a dollar, people giving you a "free" bracelet and then asked for money once they've gotten it on your arm, and people offering to take your photo with your camera and then expecting payment. But it's not dangerous, just annoying.
Barb