I'll be visiting San Francisco for 3 days. I will use BART to commute from/to the airport and also to/from Berkeley. From what I found out, it's easiest to get a clipper card on arrival. But can I use it to pay for SF buses as well? If not, is there any analogue of the clipper card for buses? Also, I was looking for something like a universal 3-day pass for both buses and BART, but haven't found any (only Muni 3-day pass, but it's not valid for buses).
The Clipper home page shows all the Bay Area transit agencies that accept Clipper; it is pretty much all of them. SF Muni is there (though you could be forgiven for not recognizing their logo as the stylized word MUNI).
AFAIK, the Muni three day visitor passports do include bus; on that page, "Muni" refers to bus service and "Muni Metro" is light rail. However, they are overpriced because they also include cable car, whose fares are designed to extract money from tourists. You will probably be better off paying per ride with your Clipper card.
BART passes do not exist (*). Their fare structure is based on everybody paying per trip, where the fare varies by the length of the journey. So just load enough money on your Clipper card and pay your BART fares as you go.
( * I know of only one exception: a monthly Muni "A" pass also functions as an unlimited BART pass, but only for trips entirely within the City of San Francisco: Embarcadero to Balboa Park. All trips that go outside the city cost the normal fare.)
Nate Eldredge's answer mainly has you covered, but there's one other option you can consider: the Muni Day Pass.
It costs $5 and covers unlimited Muni travel (buses, light rail, and historic streetcars, but not cable cars, within San Francisco), but not any rides on BART, so you'd still be paying separately to go to/from Berkeley). While the cable cars are lovely, the entire city is quite accessible without them. The day pass is only available from the MuniMobile app and not with a Clipper card.
Since the standard adult fare is $2.50 (good for up to 2 hours of travel), the day pass will save you money if you plan to take more than two rides within San Francisco throughout the day.
The visitor passports are $12 and $29 for 1 day and 3 day passes (if purchased through the MuniMobile app). This gets you the same thing as the day pass (including buses), plus cable car rides, which is why it's so much more expensive.