SFO (San Francisco airport):

BART (Bay Area Rapid Transit) can take you straight from the airport to Downtown Berkeley. The yellow and red lines go to and from the SFO airport. If your accommodations are in Downtown or North Berkeley, you will take the Red line. If you’re staying closer to Rockridge, the yellow line will be more convenient. See this trip planner for more information. Cost: $9-12.
Taxis are expensive but they do exist. Cost: $140-180.
Rideshare apps like Uber and Lyft charge around $50 for rides from SFO, although price can vary depending on time of day.
Rental cars: see this list of rental car agencies operating out of the SFO airport.
Shuttle: From SFO to Berkeley, ABC shuttle service (max 6 people) charges around $119 and from SFO to The Graduate Hotel. Airport Express (9-person van) charges around $170. This option might serve those traveling in a larger group, who can share the costs.

OAK (Oakland airport):

If you want to take BART, you’ll take the Beige line and then transfer at Coliseum to the Orange line for the Downtown and North Berkeley stops. Cost: about $10.
Taxis are $85-110.
Uber and Lyft range from $30, usually.
Rental car information here.
ABC shuttle service $89

Once you get into Berkeley/North Oakland, transportation by bus is very simple. A local single ride is $2.50