If you don't want to rent a car, I suggest then staying in South Beach, you can actually walk to a lot of places or take a cab but keep in mind that if someone is coming to see you from downtown or anywhere else, the traffic is nuts right now due to New Years and season, plus finding parking unless they valet park somewhere is nearly impossible. One more thing to consider, hotels right now in South Beach are ridiculously expensive, perhaps you can find better deals around the airport or even downtown for which you will definitely need a car. We have "car2go" in Miami,  a great alternative than regular car renting, and Über is coming soon also. Miami is a really big city but if you research accordingly, you will find a great place to stay!
Good luck to you!!!