Haven't flown in to SA yet, but lived there a bit in the early 90's... We went on a mini vacation a few years ago, and stayed at a B&B in the King William historic district called
Brackenridge House. We liked it. (My better half is Vegan, and they made special food just for her.)
It is a bit of walk from the main River Walk, but that was intentional... (Two blocks to the river, and then about a mile North via the river walkway to the main loop.) Some parts of SA still seem a little sketchy, but we did that walk at night without much worry. There are also some nice hotels right on the river too, if you would rather be right in the action.
One nice thing about SA, is that beyond excellent Mexican food, they have some really cool new and fusion type dishes. One of our favorites was
Soluna. Also at a random River Walk place we ordered awesome interpretation of "queso" that involved a slab of roasted cheese.
When walking the river, if you spot the Hyatt Regency, follow the water flowing all the way through the middle of it, and you'll pop out at street level in front of the Alamo. ...and also the
Botanical Gardens and the like are nice, if you are interested in that kind of thing.
Have fun!