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You can drive, take a cab, there are buses, trains, probably even helicopters, but what most people use and which is designated for travellers going downtown is the BART.
You have to cross the airport by foot or in the mini train shuttle (it comes every few minutes) to access the BART station. Once you get there, the price for a one-way ticket is 8.65USD to go downtown. The downtown stations are Montgomery St and Powell St, but depending where you are going you might want to figure the closest station for you (most likely between 24th St Mission and Embarcadero). The BART trains come every 8 to 20 minutes depending on the time of the day, until midnight.
It is made for travellers as well as commuters so the BART station is directly integrated into the airport and there is usually space for luggage onboard, even the largest pieces (but no specific luggage racks AFAIK).
Upvote:3
Another option, intermediate between BART and taxi, are shared-ride shuttles. These are companies that operate large vans. You board the van at the airport along with several other random travelers who are going to the same general vicinity as you. The van driver chooses an efficient route to drop everyone off at their destination in sequence. You get door-to-door service and easy luggage handling for a cheaper price than a taxi.
The main disadvantage is that it is slower: you have to wait at the airport until your van is full (of course if nobody else is going to your area, eventually they will just take you by yourself), and then ride along while other passengers are dropped off before you.
Also, the fares for these shuttles are usually per person; if you have 3 or more people in your party, a taxi may be cheaper.
Upvote:3
This is an old post and the OP likely does not need this information anymore. However, others Googling for similar results might... so I thought I'd share this.
If you are interested in going downtown directly, are arriving in SFO during morning weekday rush hours, and would like to save a few dollars, the Samtrans KX express bus is also an option for $4. (It is not nearly as frequent as BART which runs for most hours of the day at 15 minute intervals, however. So you should check the schedules and decide whether it is worth the wait.)
http://www.samtrans.com/schedulesandmaps/timetables/KX.html
From the timetables, you can see that there are no stops between the airport and downtown SF. Depending on traffic, it can be much faster than taking BART. Moreover, it's less than half the cost.
There is definitely not as much space as a BART train for luggage, but they did start allowing luggage again as of 2013: http://archives.sfexaminer.com/sanfrancisco/samtrans-bus-to-sfo-to-allows-luggage-again/Content?oid=2337301
Some more information regarding the route from local riders may be available on this Yelp review page of the bus: https://www.yelp.com/biz/samtrans-kx-san-carlos
tl;dr: The SamTrans KX express bus provides nonstop service to downtown SF for $4 if you arrive in SFO at the right time and don't have too much luggage. It can be faster than BART if traffic is not too bad.