The Tremont Hotel, a 3 1/2 star hotel, is 4 blocks from the cruise pier. It backs up to The Strand. According to one website, they have parking for $10/day. I have never stayed there, but have visited it frequently during Mardi Gras and Dickens on the Strand (see below.) It is a beautiful, restored period hotel.
You should get to Galveston early the day before you sail so you have plenty of time to explore the historic district of Galveston. It is really delightful, and the historic Victorian homes tour is fabulous. At the end of the Strand, maybe 3 blocks from the back of the Tremont, is a railroad museum that is surprisingly good. And the Strand itself, and surrounding blocks (about 10 in all) has plenty of great shopping and restaurants. The beach side of Galveston is not worth exploring, especially if you are off to the Caribbean.
And for anyone that is leaving the first weekend of December, you cannot miss Dickens on the Strand. It is a 3 day festival with costumes, street vendors, and entertainers, all in Victorian England theme. The locals go all out. During the day it is a family affair, with jugglers, parades, etc. At night, it turns into the biggest block party you have ever seen (except maybe Mardi Gras!)