Even a direct redistribution scheme wouldn't do much, because the rich will still be able to outbid the poor for the scarce apartments. Short of rich people deciding they don't want to take that job in the Bay Area or government straight up banning them from moving there, I don't think there's much anyone can do to help prevent the poor from being displaced.
Getting SF to allow new construction would help a lot (there are probably thousands of developers that would love to build 100+ unit luxury complexes in the Mission), but at this point it seems that's something of a losing battle. Weirdly, it's the very people that are protesting the current influx of tech workers and rising rents that most viciously oppose new high-occupancy construction in the "true" SF neighborhoods because it would destroy their character, so it's a tough nut to crack.