Well, it depends on what is important to you. Do you want to own a home? How about a car? Are you worried about school districts for when your kid is older? Does your SO need a job in the area?
I spent several years living and working in both areas (Berkeley 3 years, Palo Alto 1 year), with an SO who is a grad student, so I am familiar with the schools and cities. I would say first off that overall Stanford has a lot more money, and their campus definitely reflects that. However, I think it's easier to get around (both in the town itself and in the area more generally) when you live in Berkeley. I would think that cost of living would be slightly lower in Berkeley, but that's really just a guess. I think both places offer family housing, if that's something you want to look into.
Generally my impression as a non-native is that PA is a little "glitzier" and Berkeley is more understated. But you can get out of them whatever you want!