There is a lot of art, but art museums are not necessarily kid-friendly, per se. We have definitely gotten yelled at when our little children touched sculptures! The kids really like the armor and the modern art on the 3rd floor, and I like the impressionist paintings. A guide once told us that there are kids' activities Sunday afternoons, all free.
The Saint Louis Art Museum is a beautiful building that houses beautiful art. My husband and I enjoyed our visit tremendously. My two kids however were less impressed. Not the fault of the museum, they just don't cater to kids. When I go back I'll get a babysitter and just enjoy the art in the company of adults.
The Saint Louis Art Museum is located in Forest Park and is across the street from the St. Louis Zoo. The gallery is very nice and like most attractions in Forest Park admission is FREE. This is a good place to check out some masterpieces. If you go on a Sunday, that is their family day and they generally have some sort of hands-on art project planned in the afternoon.