I'm sure that someone else can recommend specific hotels, but I lived there for a long time and there were more hotels near the port than near the airport. Many of them have free shuttles airport to hotel and the port is a very short cab ride from the hotel.
La Quinta Inn/ Miami Airport was a nice stay for us and they had a complimentrary shuttle.
Carnival Liberty New Year's Eve 2007
Liberty of the Seas 5/2008+11/2009
Allure 1/16/ 2011
Vision OTS 2-14-2015
Booked Allure of the Seas... 2 / 21 / 16 *Last Gang Cruise*
Booked Celebrity Equinox ... 4 / 4 / 16 *Anniversary cruise *