Yup, they do. Most of the Google engineers I know who live in the Mission have been mugged or had their car/house broken into at least once. You just learn to not carry a lot of money or expensive stuff with you.
The reason I say it's classist is that because you have both millionaires and people who are dead broke in the same space, people have invented a lot of subtle social signaling cues to understand just who they are dealing with. Like when my wife & I go house hunting, every realtor asks "Where do you work?", and we answer "Los Altos" and "from home", and then my wife says afterwards "I think they meant which company do you work at?" Or the time I met a black dude on Muni who had a bet with his girlfriend that he could identify techies by sight, because he asked me "Excuse me. Are you a techie?" and then got it right (apparently it was the Columbia fleece that gave it away, which is odd to me because it cost me < $20 off Amazon). Or the realtor who first took me around when I moved out here, showed me a kindy dumpy midtown Palo Alto apartment, then said "It's got a Palo Alto zip. That's important to some people."
Undoubtedly this happens in the rest of the U.S, but it's different in the Bay Area, precisely because wealthy and poor live in such constrained spaces. Many of the social signals of wealth in the rest of the U.S. don't apply here because they're stupid - if you park your Lambo in the Mission, it will get taken for a joyride, if you wear an Armani handbag it will get lifted from your car, and there's no room for your gated community outside of Atherton. Our billionaires are too busy coding, raising capital, rollerblading, and kiteboarding to worry about things like their wardrobe or car. You get startup founders who own companies worth tens of millions and yet use a 5-year-old iPhone with a cracked screen.