Apologies for resurrecting the veer off topic...but we sailed Oasis this past summer and left the port area in Nassau. On Oasis's 9-nighter, it was Emancipation Day when we arrived in Nassau, so it derailed some of our plans (the pirate museum, for instance, was closed).
We walked over to the Queens' Staircase, then down to the Bahamas Rum Cake Factory. Never felt unsafe. The infrastructure is in rough shape (sidewalks were more like stepping stones in many places), but that's not surprising.
We do live along the Jersey Shore, about an hour from NYC. I worked in Manhattan for 10 years and we still go up to the city a few times a year. Hard to compare the two, but I feel like general safety awareness is the same for either.