Being "progressive" is great, but we live in the real world and if you refuse to do things to lower prices in the name of culture, prices will keep going up. Ironically, this will also hurt, if not destroy, the "culture" that they're trying to protect in the first place.
San Francisco is statistically the second most dense city in the US, wedged into a geographically bounded ~50 sq miles. It's dumbfounding that there were probably more high rise apartment buildings (>20 stories) in my zip code back in NY than in all of SF.
This is how you solve it:
1. Allow development, stop getting in the way.
2. Create punitive pigovian long term unused property taxes to encourage property use. Stop land owners from letting property just sit there unused and discourage using property just as a fancy bank account for the wealthy.
Raise Taxes
If you want to lower prices and you can't/won't increase supply, the obvious solution is to lower demand. Adding a sharp jump in the local income tax at a high income level will free up housing space as wealthy citizens leave the city. The less affluent won't see a tax increase, since it's outside their bracket, but they will see housing become available, lowering rent prices.
Of course, this will be a hit on the economy as all the wealthy individuals leave the area, but it decreases rent while keeping culture constant.
Build vertically. Look at Tokyo if you need ideas of super density.