Crime in Miami IS BAD. Regardless of what anyone tells you I can honestly tell you that it is bad. All you have to do is visit the miamiherald.com on daily basis and see for yourself. I was raised here btw, then we moved out for a few years and recently returned. Up on our arrival, within 3 months, our new home (with alarm) was burglarized (this is the second time my home is burglarized living in Miami). Almost everyone I know has had their home burglarized. I can't even keep count how many times my cars have been burglarized or how many times I've seen police search helicopters hovering on top. I can go on with many stories. And I've only lived in "nice" upscaled neighborhoods.
As far raising children, well I would say Miami is near the bottom of the list. In my opinion, this is not one of the better places to raise a family (if you can avoid it that is). Of course this all depends on what you consider decent and what your standards are and what schools you want for your children and what type of friends you want your children to make. I went to school here, so I know. And things are worst now than 20 years ago.
You say that you're on a "budget", well unfortunately you will not be able to settle in anything I would consider decent in Miami. Of course this can subjective. If I had to stay in the Miami area, the only places that I would consider are not very affordable unfortunately. This includes "certain" areas in Coral Gables, "some" communities in the Pinecrest, and maybe some in Palmetto Bay. Of course, then you have the very high end areas like Gables by the sea, Gables estates, Cocoplum, etc.
Remember that there are many things that make up a city, things like level of education among residents, quality of schools, and the type of people for example will have a big influence on crime and the standard of living.