I grew up in museums as a child. My dad was in the military so we traveled all over, and we lived in Europe for several years. We explored places like the Louvre (the Mona Lisa is tiny), the Palace of Versailles (almost got kicked out for sitting on some old chairs), perfume factories, glass blowing facilities, old castles, zoos, planetariums, art museums, natural history museums, science museums, the list went on and on...and my sister and I complained the WHOLE time. "This is BORING! We want to go home! Why did you bring us here??? We want to go to Disneyland!" And over and over they told us, "Someday you'll appreciate this, damnit!" So now, as a parent, I realize the value in what they were trying instill in their unwilling, wretched children: Art and history and exploration is COOL! And Disney World, while entertaining, doesn't even compare to the real thing...
If you have relatively well behaved kids, I don't see why parents can't feel welcome to take them into "grown-up" museums. Docents following around kindergartners, begging them not to touch anything, are underestimating kid abilities and self-control. There is real intrinsic value in learning to appreciate creativity and imagination, and kids today are the ones who will be supporting these museums in 30 years!
The San Jose Museum of Art is a great kid resource in the south bay. It's not too big, they usually have a wide selection of different art mediums to look at, and the ticket admissions are more than reasonably priced. They also offer youth and summer programs that look like a lot of fun!
I've taken Matthew to see the M.C. Escher and Op Art exhibits in the past, and was recently there for an event opening of the Frida Khalo exhibit.
The DeYoung Fine Art is also a great resource for art exploration. The food is good, the art is plentiful and varied, and the Music Concourse is a great place to get some energy out after being indoors.