You might have read comments about how great Belize is (or certain parts of Belize), but I doubt anyone said that Belize City was a great place. I have lived there.
Belize City is divided by a river into the North Side and the South Side. Most of the crime occurs on the South Side. I'd stay away from that area if possible, and especially at night.
So far this year, there have been 35 murders in the City (70 in the whole country), and at least that many shootings (none of the victims have been foreigners).
The North Side has some better neighbourhoods (e.g. King's Park), but while there aren't as many murders, home break-ins occur -- that explains the burglar bars on the windows, high fences with broken glass, security cameras and watchdogs.
Better neighbourhoods (Belama) are located a mile or so out along the Northern Highway.
The downtown area is generally safe in the daytime, but I wouldn't walk off down streets that look deserted.
PM me if you need to know anything more.