The problem is that the fallback position tends to be building no new housing at all. This results in the same new people with more money moving in, driving up housing costs and gentrifying the area. No amount of not building housing makes rents low.
What is that mistake? Permitting new, nice housing?
If this is your opinion, I cannot agree with it.
Firstly, something like 10->20% of all new residential construction must either be set aside for "affordable housing", or enough money must be given to The City to construct those units on behalf of the developer. Secondly, San Francisco's mistake is to -through a variety of screwups- make it nearly impossible to build new, dense residential buildings in the city.
If a district is able to build enough housing to meet current and expected near-future demand, rents will usually remain stable. See footnotes 0 and 1 at [0] for some info on long-term trends in the area.
edit: There is an alternative and that is to find a second hand contracts where you will have to move out in a year because most co-op boards won't allow the owner to rent out for more than a year.
Also, if you've been in your current apartment for 5+ years, changing apartments effectively means double rent for you. Everybody I know who has an OK apartment in Berlin stays there, because an extra room etc. would mean 500-1000 Euros rent increase.
Have you been following the events surrounding the problems with rising rents in San Francisco? What you're describing is what has already happened here. I cannot overstate the importance of avoiding the mistakes of SF. Refusing to build in a place people want to live in will not lower rents. It only drives gentrification, evictions, and higher rents.