I live in an unfashionable neighborhood south of 280, but even here, you'd be lucky to get a decent 3br for less than a million. So assuming you have a down payment together, you're probably looking at 4k a month in mortgage. You get a tax break, yeah, but you also get property taxes, maintenance, and insurance, so it's a wash.
Child care is the real killer. Standard 8am-6pm child care runs north of $2,500 a month. Same for preschool. If you have two kids, oh man. So now you're looking at
6,500 a month with one kid 8,000 a month with two kids
Keep in mind, if you can afford the child care in SF, odds are you're too high income to qualify for the child care tax break. So all in all, it really is true that it isn't easy to make it on two incomes totaling 200k a year.
Beyond that, though? If you're north of 350k, I'd say this is about expectations. You should be able to make it fine.
There is one strategy - remain a one income family! $120k a year for a family of 4 qualifies for government housing assistance in SF. Developers must set aside a number of units for "middle". It's hard to get, but if you can pull it off, you're guaranteed a lower rent (I think it's about $2500 a month at that income level) and it's intensely rent controlled.
So Silicon Valley employers, take note. Hire a middle aged person with a family for $120k a year, and you can get that highly skilled worker in critical shortage that you've told congress you can't hire at any price, and you can get the taxpayer to cover the gap between the salary you pay and the housing costs in the Bay Area! Seriously, consider the benefits of hiring older people with families.