We were just in Galveston last month with our two year old. We stayed at the Holiday Inn Club Vacations Beach Resort: tripadvisor.com/Hotel_Review-g55879-d564382-…
It's a little ways out of town (5-10 minutes) but it is literally right on the beach. They are little mini condos. Ours was a one bedroom (with a door so we could put her down and still sit up and watch tv or whatever in the evenings) and a fold out couch. It had a little kitchen area complete with all plates/pots/pans, fridge, stove top, microwave and dishwasher. It had a washer and dryer unit in a little joint hallway that was shared with another unit. Only downside is there isn't any room cleaning during your stay so if you run out of towels you have to do a load of laundry but they give you the detergent for the machine.
Like I said it's a little ways out of town so to eat you either need to stock your fridge (we stocked breakfast and lunch items - there is a Wal-Mart and Kroger right on Seawall that you have to pass to get to the hotel) or drive into town (we did that for one big meal in the evening). The hotel has a decent sized pool with a little splash shelf for little ones to play on. Then there was walkway down to the beach. The beach is shared with a island public park so there is a life guard on duty and people other than hotel guests can use it. But since its so far down the island it doesn't get as busy as Stewart's Beach in town. They also had a nice outdoor shower right before you reached the pool area so you could rinse off and drag a little less sand back to your room. They had a few picnic tables and grills on the property as well.
One last plus all rooms have a view of the gulf and a small balcony. We loved this place.