I really don't have a recommendation for you regarding where to stay in Galveston. I live nearby, and the only times I stay overnight in Galveston is for a special occasion and we normally stay at the San Luis or Moody Gardens. Both of the facilities are very nice, but also a little on the pricey side.
These are the threads that came us when I used the search feature of the CM board. This has been discussed often. Read through the threads and I bet you can find some information that will help you.
If no one in your party is a smoker the Commodore on the Beach is a nice place to stay. It's an older hotel, but clean and all the rooms face the Seawall.
For others driving in, they have free cruise parking. It's not "covered or absolutely secure", but there's a camera in the lobby that looks over the parking lot. (We stayed there twice and never had a problem with our car.)
Many people stay at the Galvez Hotel, the Tremont and the Harbor House.
As someone else said, check out www.galveston.com. Personally, I'd check the prices and info with each hotel instead of booking through the Galveston.com website.
Galveston is a very small city, so wherever you stay will be close to the port.
"A truly happy person is one who can enjoy the scenery on a detour."