Look at housing in a part of the city where you want to live. By that, I mean safety on the street and schools for your kid. $4000/month is a dirty hippy pad in a crappy part of town. Fine for a single guy, but would you wife want to lug a kid around dirty streets and aggressive homeless people? Also, forget having a car if you rent in the city. Parking is a minimum of $350/month.
Also, look at the schools. They are ridiculously bad, except for the magnet schools, which are some of the best in the nation.
Finally, figure in the state income tax. It's quite progressive, and tops out at 13%, on top of Federal tax.
You could live in the East Bay, but the commute to the city is horrendous, 30+ minutes on BART, plus commute to the station, and then from the station to your office. It'll probably be more than an hour each way.
In a previous thread where a guy took a job on the Peninsula, my advice to the guy was that unless you're making $300k or more, renege on the job offer and stay where you are. If you have the cash, the SF Bay Area is paradise. Wonderful weather, beautiful scenery, plenty of culture and entertainment, world class schools and universities, and short commutes. If you're on the wrong side of the divide, it's hell on earth. HORRIBLE schools, high taxes, long commutes, extremely high cost of living. Unless you moved here in 1988 and bought a place, you need to make $$$ to do okay here.
People from other parts of the country have NO IDEA how expensive it is. It's closer to Manhattan prices. Nothing else in the country compares to how hard it is for regular professional folks to live here, if you have a family.