And is there public transportation to San Francisco from San Jose airport?

Sort of. You can take CalTrain to the SJ light rail and then take a shuttle
bus to the airport (possibly there is a superior alternative involving more
bus and less rail). I sure wouldn't bother.

Take the same bus, the VTA #10 airport flyer, in the other direction
from the airport directly to the Santa Clara train station. The bus
stops every 15 minutes at terminal A and takes 20 minutes to get to
the train station.

You can take any of several door-to-door shuttle vans to San Francisco
for about $70, or you can take the Airporter bus to SFO for $15 and
take BART from there.

I agree that it's all rather slow ways to get to San Francisco and I
wouldn't bother either. I'm flying into SFO tomorrow and I'll take
BART.