I know several hotels in downtown Oakland close to BART (absent another strike) and therefore the City. The Oakland Museum of California is excellent. There are multiple excellent places to dine. There is for most of the hotels downtown proximity to Oakland's Chinatown and its does have its merits being less touristy than San Francisco's. But few on the California forum suggest it to budget travelers.
Maybe Rockridge rather than downtown for those who are not prepared for an inner city experience, also close to BART, and also full of interesting things to do and also excellent dining places.
I think we do a disservice to travelers when we neglect those parts of the Bay Area which have merit, not just San Francisco. I have been troubled here before when people have been snide about staying outside the City when it is not known why the traveler might be doing that.
Sorry but I would not suggest that a traveler stay in San Jose. It is a bit to sprawled for my taste, but that is different.