Belize has a particular reputation lately that it is dangerous, but that is only in certain parts of the country. Is Belize City one of those parts?
Short answer: Yes. Does that mean you will be a victim of a serious crime if you visit the city? No, not at all, Many people visit the city every day without any issues. Instead, it's usually specific areas that are more crime-ridden. The south-side streets of the city away from tourist centers is where most of the more severe crime happens. The north side of the city and the main streets downtown are usually pretty safe; especially during the day.
However, it is important to remain aware of your surroundings and be very careful about traveling at night. It's not recommended to travel at night, and even for short distances, it is advisable to spend the money to get a taxi to drive you. A good thing, if you can call it that, is that fortunately for tourists, a lot of the homicides and crime are gang-related and these crimes tend to affect tourists less. The type of crime that affects tourists the most is pickpocketing, which can be largely prevented through good awareness of your personal property.
In short, yes, Belize City is more dangerous then most of Belize, but if you stay cautious, it is easy to avoid getting in any trouble.