I asked the same question of a friend who just moved from Miami since we are on the Carnival Liberty at the end of May. She said the area right around the port is pretty risky. She suggested staying in the South Beach area. It is just across the bay from the port.
Also, from what I read on some of the boards, the free transportation from some of the hotels is really crazy. Many folks end up just taking a cab due to the long wait times.
Have fun on your cruise!