SF Jazz Center
I would definitely suggest the SF Jazz Center. While shows are "all-ages", they have special family programming too (like Family Matinee, where the performer of the week/weekend plays a special afternoon show for children/families on Sunday's, I think. The musicians play the music they would normally play during their nighttime shows, but there is also engagement with the kids in the audience): https://www.sfjazz.org/education/kids-and-family/ I have also been to Thursday-Saturday night shows and have seen several kids scattered here & there in the audience. I had SF Jazz membership for two years, but didn't renew it last year because I had a baby. When she gets older, we'll start going to shows again.
Also, check out Ashkenaz, an all-ages world music venue in Berkeley: http://www.ashkenaz.com