Belize city is definitely dirty and scuzzy. I didn't feel it was all that dangerous though. I walked around at night mostly around the areas where the ferries leave from and didn't feel threatened. The murder rate is high, but if you aren't in the drug business its very unlikely to affect you. I bought a vehicle there and drove around all over the city. No problems with traffic, etc.
Keep in mind that even if you are working in Belize City you don't have to live in the city itself. There are only around 50,000 residents so you could live near the airport and commute into the city in under half an hour I imagine. If you bought a decent boat you could potentially live on Caye Caulker and boat into work.
I think its a 30 minute boat ride.
I really liked Belize and I think it would be a decent place to live depending on your personality. Do you like the small town vibe? The entire country has around 300,000 people I believe, so everywhere has a very small town feel. I felt like the majority of people I met there were friendly, and strangers are more likely to randomly make a comment/greeting to you than in most other places.
Keep in mind that the number of services/activities is limited due to the small population. If you like hustle and bustle you may end up bored.