I've lived in both SF and NYC and feel like Oakland has much more of a stigma than Brooklyn does simply because of transit access. You can live in SF and enjoy walkable local communities. You can live in Oakland and enjoy walkable local communities. But getting between the two can be a much more arduous process via Bart or bus than the many plentiful subway connections between Manhattan and Brooklyn. Visiting friends, of even the prospect of a commute, between SF and Oakland would have given me much more pause than getting over to Brooklyn. BART and the transbay buses are woefully inadequate for an area which is trying to pitch itself as a region rather than two distinct cities with a bridge between.
Not only are train lines between Manhattan and Brooklyn much more plentiful, they fan out to a much larger collection of destinations on either end. It can be a one-system trip, which makes a big difference.
BART isn't bad by any means, and SFMTA/AC Transit do a decent job, too. I've lived in Miami and around Tampa, two other large cities which do an almost immeasurably worse job with transit. But BART is still a very commuter-centric network for getting commuters from suburbs to downtowns, and isn't a substitute for a true local subway network. Thats where the stigma comes from, in my opinion.
Yes, but that's all you can do. If you're in pac heights or the presidio or ... anywhere west ... you're going nowhere.
So, yes, you'd prefer to live in SF than Oakland but 15 years or so ago, you'd probably have preferred to live in Manhattan than Brooklyn.