I live in Oakland, and take the BART (from MacArthur Bart Station) everyday to commute to SF and walk home from it late at night.
I too was scared of living in Oakland but it's actually a really chill city and I find it much safer than San Francisco, I would probably not go to San Francisco if it wasn't for work as there is nothing Oakland can't provide. I love Berkeley, not sure what's the bad rumours regarding Berkeley are about because it feels really safe, it's very much a Uni town though, were the average population age is probably early 20's.
120k to 160k is good enough for two people without dependents but if you are going to live in places like Alameda, Walnut Creek or Lafayette, you are quickly going to realize that you will need a car so you don't get bored senseless, as public transportation in those places is limited and transiting from the Bart from those can be a pain, especially during rush hour.
BART is alright, I don't feel more scared in it than I would other places but yes, there is crime and stuff happening sometimes but it's everywhere in the US, crime is a thing in the US and people getting stab or shot are common news (not necessarily in the Bart, but yes sometimes!). Cars break in are rampant in the Bay though, so that's something you need to be aware of.
The stories about crime are grossly exagerated, I live in Oakland, supposedly a scary place full of crime, but it's lovely, we have a beautiful lake and I find it much more laid back than SF and find the quality of life much better, I walk everywhere (though mind you without a car distances are long and public transportation not as good as it could be!) at anytime of the day/night and don't fear for my life.
You can definitely survive on 120k+ in the Bay, you can also probably afford to live closer to work and I would, you will waste so much money in Uber and public transit commuting from an hour away when Oakland, Berkeley, Piedmont or even SF, probably have more to offer for a couple in your age range.