In the cities and the mainland pretty much I would pick another sightseeing trip. The organized trips to the archeological sites are still okay, but in all things, mit would depend upon your attitude, capabilities, and personal risk assessments.
On the Bay Islands, the only issues that I would really concern myself with would be on Roatan. The other islands have an issue of compactness of population center that pretty well limits many exposures. Roatan allows you the possibility of getting pretty far afield.
To that point- the vast majority of visitors to Roatan go to resorts to dive (at least in this board). I encourage them to stay at the resorts in the evening party hours. Maybe it's because I have no toilerance for hung-over divers, or maybe it's just cause I don't want to have to help them out of a jam.
Personal crime to the visitor has increased for the unwary. Why any visitor inists on driving their own car there and having to deal with the (RARE!!!) occasions of banditos, much less quirky driving laws and loiability concerns- you got me.
So I speak of Roatan- Stay in a touristy area. If you must drink, try not to be a "soft target".
Don't fear, but don't be so innocent that you have no fear.