From one Mother to another -
I always go a day early when sailing out of Miami and have stayed at the Best Western Bayside Hotel - multi story inside rooms (3 Stars by AAA) within a few blocks of the port. There is no shuttle service to the port...but it's a very inexpensive cab ride. It's been several years...but this hotel was much nicer than a typical Best Western although a little costlier - it was still a reasonable price and a short cab ride from the cruise pier (about 5 blocks away). You can walk across the street to the "Bayside"....a beautiful indoor/outdoor shopping mall with many restaurants (Bubba Gump Shrimp, HardRock and many many more), free outdoor concerts at night w/dance area...lots of local entertainers outdoors at night and nice tours of the bay and celebrity homes. There are horsedrawn carriage rides around the area in the evenings as well. It's really an awesome location. I loved the Bayside hotel experience and the activites across the street....as much as the cruise.
Take a taxi from the airport to the hotel and where the doorman will meet you and assist in unloading your bags.