If you have a car, go to the Berkeley science (and math) museum up in the hills above campus, beyond Livermore Berkeley Labs. [It's awkward and takes a lot of time from the BART station as you might have a long wait for the right shuttle.] You could combine this with a swim at the Strawberry Canyon pool, maybe climb to the top of the campanile on campus and admire some of the architecture and vegetation.

Are you planning to do Muir Woods or go on one of the drives near the big trees?

I like the drive over the Golden Gate Bridge (IIRC naturally beyond the park with the Japanese museum and garden) to Sausalito (art galleries, so unlikely to interest kids much, but you might find a show with work by Japanese or Japanese-American artists) or up the coast, maybe stopping at a beach (but the water is too cold to swim). There's a small park at the base of the bridge that I think has some information about how the bridge was built. You can also walk or bike over the bridge, but it might be better to just do a bit in order to sense how big and high it is and come back to the car.