B.L.
Don't get me wrong, I love Galveston and have made the trip many times with friends. That being said to me the clear choice is San Antonio! Galveston beach is ok but it will be SO hot in July your baby will be miserable. At the age of 1 he will not fully be able to appreciate and play in the sand and surf. He may just hate it altogether. The sand will be HOT and the salt water will be sticky and might give him diahrea. That's not to mention the prep and clean up. We went when my brother was 1 and my mom ended up spending the whole time in the room with him.
San Antonio has the wonderful river walk you can stroll down. He will enjoy the river boat, The zoo and Sea World not to mention the hotel pool.
I think if you had an older child I would choose the beach but I personally I would not take a 1 year old to the beach just yet.