From San Francisco Airport: take a BART train, about $5.00 to the Downtown Berkeley station. The BART station is in the airport through a single doorway from the international terminal. The ticket machines accept credit cards. Berkeley is on the Richmond line, and it may be necessary to change trains en route. Usually these are programmed changes and it's easy. Take the first train that leaves the airport.
From Oakland Airport: take the shuttle bus to BART, $2.00 exact change (credit cards OK). Take a Richmond line train to downtown Berkeley (about $2.00).
Other means of transport: Both airports have door-to-door shuttle services, which are shared vans; this might be appropriate to a hotel not near a BART station. There will be a dispatcher at curbside to assign you to a given van. There are also taxis, of course, but they are expensive from SFO and prone to traffic jams getting across the Bay. Rental cars are convenient but parking at SSL is controlled, so be sure to request a parking permit.
By taxi: about $8.00 from Berkeley BART.
On foot: it is a hard slog, about 45 minutes via Strawberry Canyon and Centennial Way.