Alamo Beer Company is nice. Big and spacious with Hays Street Bridge right above it. Just need to watch out for homeless around the area.
https://www.alamobeer.com/ Cherrity Bar is a outdoor family friendly bar. I believe every drink you buy they will donate a portion to a charity. They pick a different charity every month. It's right across the street from the dome.
https://cherritybar.com/ The Friendly Spot
Popular hang out. They have a small playground for kids.
https://thefriendlyspot.com/Backyard on Broadway
Kid friendly bar. The SA Aggie club would have watch parties out there. A little drive away from dome but not far.
https://www.backyardonbroadway.com/Smoke BBQ
I would stay away from Smoke. It's a party bar and could be a party crowd during the day. Not sure if it's kid friendly ever since they moved to a new location.
Check out the sites or their Facebook or Instagram pages.