Your best bet for flying in is to fly to San Jose International Airport. Once there, you’ll probably want to catch a South and East Bay Shuttle or Super Shuttle to the Domain. Taxis will make the trip for about 35 to 40 dollars; you can get a shuttle service for 18 to 25 dollars.
If you fly into San Francisco, don't even think of taking a taxi to the South Bay. A shuttle is 24 to 35 dollars and about a 40 to 50 minute trip. Or you could take BART to Millbrae (via San Bruno), and transfer to Caltrain heading south to Lawrence Station, where you will have to catch a cab to The Domain. It takes about 90 minutes total, and costs about 25 bucks in all, but it’s a pretty good way to go, especially if you’re arriving during rush hour.
If you fly into Oakland, there’s a bus that’ll take you to the BART station. Then you can take BART all the way around the bay to Millbrae and follow the directions from SFO. It’s a long trip, so you’re best off flying into SFO or SJC.
The Caltrain station that’s closest to The Domain is Lawrence in Sunnyvale. It’s about a five minute cab ride from there.
AMTRAK comes into Santa Clara by Great America. This is about the same distance from The Domain as the airport. There’s also Diridon Station in Downtown San Jose, which is further out.
Greyhound will drop you in Downtown San Jose (around the corner from legendary San Jose dive-bar The Caravan). From there, pretty much your only choice would be a cab to the Domain.
From San Jose International Airport
From San Francisco Airport
From Oakland International Airport