The High Trail above Washburn Lake is a hike one could do using public transportation. Often people take the bus from Yosemite Valley to Tuolumne Meadows and hike back to the Valley using a variety of routes. This less used trail is one of my favorites. If you hike it in September its possible that you won't see anyone else.
From the San Francisco International Airport
You'll need to get on Bart (Bay Area Rapid Transit) which only goes to the International Terminal. You can walk or take the Air Train Shuttle. You'll want to take BART to the Richmond Station. This Bart train doesn't go to Richmond so you'll have to transfer at some point. This can be done any time after passing Daly City but make sure you do it before the MacArthur Station in Oakland. Once you arrive at the Richmond Bart Station walk over to the Amtrak Station and get a ticket to Merced California. See below on directions on getting from Merced to Yosemite.
From the Oakland International Airport
There are two ways to get to Amtrak. One is to buy a Two Dollar Bart ticket in the terminal or at the Air Bart stop for the bus trip. Board Air Bart to the Oakland Coliseum and board Bart. Notice that you'll have to buy another ticket for BART. You can take Bart to the Richmond station and walk over to Amtrak. The second cheaper option is to board the #58 AC Transit Bus to the Amtrak station at Jack London Square where you'll board Amtrak for Merced. See below on directions on getting from Merced to Yosemite.
From the Sacramento International Airport
From the airport take the Yolo Bus #42A to downtown Sacramento. Get off the bus at the first stop in town and walk a block north to Amtrak. You'll be put on an Amtrak Bus to Stockton where you'll board a train for Merced.
Merced to Yosemite
Currently there are three morning buses and one afternoon bus leaving Merced for Yosemite. If you want to look at the schedule look for the Highway 140 bus. The route varies slightly depending on what time of day so don't expect it to stop at every stop. If you took Amtrak to Merced the bus will leave from the station. It makes a number of stops along the way. This is convenient for people working in the park or for tourist not wanting to drive to the park and have to find a parking space. The final destination is Yosemite Lodge however, if you want to connect with the bus to the east side of the Sierras you'll want to get off at the Visitors Center. The bus doesn't always stop there so you might have to make a special request with the driver. If you plan on backpacking in the park the driver will drop you off along the road. Of course you'll need a permit first. After you're through with your hike just stand along the road when the bus comes by and wave to get him to stop.
NOTE: Unless you live in the area you'll probably be spending a night somewhere unless you take an overnight flight. Don't count on finding a place in Yosemite after you arrive. You'd be best to make arrangements ahead of your arrival. If you plan on backpacking in Yosemite you'll need a hiking permit. Permits are available at the Yosemite Valley Wilderness Center. There are some permits available on a first come first serve basis but I wouldn't count on it. Some of the more popular permits go fast and if you're not at the window when they open in the morning you could be out of luck for that day. It would be best to reserve on ahead of your arrival. Make your reservation early!