This is not true. There have only been 85 Zika cases contracted in Florida to date; Zika has only been found in two places, one neighborhood in Miami (and that neighborhood was declared "Zika free" yesterday) and Miami Beach. They are still working on Miami Beach but have eradicated a lot of Zika-carrying mosquitos there. This is per the CDC.
To answer the OP's question, I have to travel to Miami a ton for business and I really like the Courtyard Miami Homestead. It's fairly close to the Keys and central Miami, relatively inexpensive, and there are restaurants and a grocery store in walking distance. I think I paid $129 a night the last time I stayed there (in July).
I would plan on having a car if you want to see things like the Keys and the Everglades, as they're pretty far from Miami and I don't know if any public transportation that goes there. There might be, though - I always have a car through my work so if I bring someone with me we drive. I think it takes about 3 hours to get from Homestead to the end of Key West.