Your ability to find a job in New York really depends on your specific skills and work history. I will say that you will DEFINITELY have to start your search well before your arrival if you choose Columbia. The unemployment rate in NYC went up dramatically between Feb and March, and many companies are in hiring freezes. Obviously, finding employment in California will also be difficult, but the cost of living will make a huge difference, especially while you are unemployed and looking or if you choose not to work. In my opinion, the potential for an earthquake is a risk I would take considering a life in NYC with a family and no job will cost you upwards of $3,000/month (if you're lucky). If you can only work 20 hours per week, I would advise that this would do very, very little to help you.. I hate to sound gloomy, but it's an unfortunate reality that most people don't realize before moving here.
To answer one of your other questions, it will absolutely be possible to find a job anywhere in the country (including NYC) if you attend Stanford.