The Pier 39 California Welcome Center (as mentioned in this question) does offer luggage storage. It's $5/bag, 7am-9pm. I've never used the service, but it's run by the state's tourism promotion organization and I would expect it to be reasonably reliable. The catch is the location: it's Pier 39, which is an extremely touristy area, and it's not directly connected to the main public transit (BART) train that you might use to get to the Oakland airport. As such, you will spend a chunk of your day in SF traveling to/from Pier 39 instead of seeing the city. If you plan to get to the airport by taxi (which could easily cost more than $60+tip depending on traffic), this is less of a problem.
A more central location to store your luggage would be the Union Square/Powell area, which is easily reachable by BART both from SFO and on your way to the Oakland airport. There are many hotels in this area, and while some will frown on storing luggage for non-guests, you may find an accommodating bellhop, especially if you're prepared to offer a tip. Saying you'll get a drink or bite to eat at the hotel restaurant could help. If they refuse, you could try another hotel. A few years ago, the Chancellor Hotel, just off Union Square, was offering to store luggage for just a tip (thanks to vacapinta on MetaFilter for this recommendation).
Another option might be the Temporary Transbay Terminal. The Amtrak office offers luggage storage, though I'm not sure of the fee. This is a few blocks walk from the Embarcadero BART station and the Ferry Building.
Finally, you might try a museum coat check, particularly if your bags are on the smaller size and you plan to visit the museum anyway. SFMOMA (SF Museum of Modern Art) is near the Powell BART station and their website doesn't say anything about refusing luggage, just large instruments, animals, and bikes. It is, however, newly reopened after expansion and advance tickets are required to enter. The coat check downstairs might technically be accessible without a ticket though. The Asian Art Museum, near Civic Center BART, says they take purses, bags, and large parcels. The Exploratorium (interactive science museum) has coin-operated lockers inside the ticketed area. Some in my experience are large enough to accommodate at least a carry-on sized bag.
