Best way to use public transportation for an afternoon / evening in San Francisco (from Berkeley)?

Take BART from Berkeley into SF. Get off at Powell Street, which is in the heart of downtown. Once you are there, many things are within walking distance, and local bus and subway rides are $2.

As others have pointed out, finding parking in SF can be impossible and it will cost you a lot ($20+ per hour). Ditch your car and use BART. We now use Clipper cards which gets you on both BART and all Muni (the SF public transit system) conveyances (although cable car rides cost more since they're mostly for tourists). Look at the BART, Muni (SFMTA) and Clipper card websites to see if they have some day passes or tourist passes because Clipper cards cost something like $5 just for the card alone, and then you have to load them with dollars so you can use them on BART/Muni--and you don't get a refund if you don't use it all, if you run out of money but ride the transit anyway you could get a hefty fine (last I heard it was something like $275 per person per violation) and "I was going to load more money on the card after I got off the train/bus" or "I didn't know I'd used up the value on the card" doesn't fly. Your family needs to decide what they want to do in SF--because there are double-decker tour buses you can pay for a ride on that will deal with SF traffic (and we also have "no left turn" issues) and will let you sightsee many of the major tourtist places (again, you can do a web search). There is also this "hop on, hop off" bus (that keep getting in my way) where I think you just buy a day pass and you can get on any one you want, get off where you want, and then get on another one that comes by later all day long--but reserach that too since I live here and don't use this. But your family firsts needs to list what they hope to see/do in the city, then you can better plan routes and means since you don't want to be going back and fotth a whole lot.

You will pay through the nose for parking in SF. $20 for a garage if it has a spot. Take the BART unless you are going to several colleges in S.F. then you can use a car. I dont know about a day pass. Where I live there is no BART.

Parking is expensive and depending on the San Francisco neighborhood, can be difficult to find. BART is a good option and once in SF, Muni has trains, streetcars, and buses that can help you get around. If you know what stations you need to go to on BART, you could just buy a round trip ticket for each passenger (you can't just buy one ticket for the entire family since each passenger needs their own ticket). My aunts, cousins, and grandmother live in different parts of San Francisco and I often have to park a mile away and walk to visit them when I drive.