Just took our 4 year old down to SA in January and visited Sea World. Stayed on the Riverwalk so we could see the Christmas lights and just to walk around a little bit. There's not much really cool stuff for kids (or most people) on the Riverwalk. They have some of those bootleg attractions like the little aquarium or the Ripley's spot. I guess kids get a kick out of that stuff regardless... Our kid liked our morning stroll along the Riverwalk and said it's "so, so beautiful." We took her to the Alamo for a quick picture and to confirm her as a Texan.
But you're walking distance to Hemisfair park which has a really cool playground on the west side (Yanaguana Garden). And if you're there, you might as well continue walking to Rosarios.
Grant Pinkerton's flagship barbecue spot is now downtown too, so if you need meat, that's your spot.
The light show on San Fernando is really cool:
https://www.mainplaza.org/san-antonio-the-saga/Get on a boat at the Riverwalk. Even if it's just the shuttle, it'll be fun for the kids. Take it up to the Pearl District if you want to hit the food hall and the park/splash pad there. Check to make sure they're running though. When we were there last, they were still offline, with the exception of tours and speciality cruises.
You've got the Zoo and the Japanese Gardens in Brackenridge Park. And the Doseum is right down the road, which is probably the best kid's destination in town.
If you stay on the Riverwalk, park at the Convention Center garage (41 Bowie St, San Antonio, TX 78205), which should be $10 a day and will save you some $$.