Belize City is similar to many places in Central America. I never felt unsafe during the day. Even in the evening, using a taxi and going out I had no problems. As with any poorish country, don’t flaunt jewelry or valuables. Leave most cash and valuables in your hotel safety box if you’re going out.
The ferry was a fun experience. I don’t recall the name of it, but it was fairly quick and not expensive. Seeing locals take the trip was fun, too. I think they went every couple of hours, but I’m sure you could google a schedule.
I don’t think this is the company I used, but is the first google hit. 9 daytime trips a day starting at 8am. $28 roundtrip for adults.
http://www.belizewatertaxi.com/book-now