None of the regionals have bases in the bay area. The closest bases are Monterey (MRY), Sacramento (SMF), and Fresno (FAT)for Skywest. I believe American Eagle has a fairly senior line on the ERJ that overnights in San Jose (SJC) nightly, LAX based.
Southwest has an Oakland base that I here is pretty junior. If you go fractional, Netjets and CitationShares has San Francisco (SFO) listed as a base or gateway city.