I lived in NYC for 4 1/2 years, and have lived in the Bay Area for about 4 1/2 months (with eight months prior in Sac). In my opinion, SF is far more kid-friendly. Many of the neighborhoods in the western part of the city are practically suburbs, with beach access and parks galore. The bus transport to the east side is fairly accessible, culture and diversity abound every which where, and you can actually have a car. Square foot by square foot, it is actually a bit cheaper (although the California taxes probably level the playing field just a little bit).
Interestingly, while it is as diverse as SF, if not moreso, the neighborhoods are actually fairly segregated by race. While interracial mingling is not entirely uncommon, the spread in SF is far more heterogenous than NYC.
In order to get anything similar in New York, you have to venture to the outer boroughs, which offer a bit more of a rough-and-tumble, working-class vibe for a very similar price tag, and a far longer commute to Manhattan than you need in order to get from the West Side to the East Side of SF.
However, I do love New York, and want to clarify a few things -
1. It is most certainly NOT "LA on an island w/ snow." The population density is far greater but traffic is slightly less blocked, although public transport, biking, and walking are by far the most common forms of travel.
2. Believe it or not, the weather is not as bad as some may think. I would say that 50% of the time (spring and fall), NYC has better weather, and SF is better off the other half of the time. Spring and fall are beautiful in NYC, with clear skies and crisp temperatures. Winter is bitterly cold, while summer can be humid and sweltering. San Francisco is cool and foggy the majority of the year, with microclimates and pockets of heat and sunshine. Depending on where you are in the city, the weather can vary greatly.
3. Cultural diversity and activity cannot be beat. It is one of the most exciting, fascinating, multicultural, historical cities in the world. The biggest downfall is the lack of outdoor activities, but even outdoorspeople can get their kicks here if they find the right places to go.
I do think that San Francisco is by far the coolest city in the United States, and a far better place to raise kids, but don't overwrite New York City, as it is a fantastic gem as well.
(Yes, I can go on forever about this topic).