Most people do Napa with a rental car.
But you can take a Bus 'Day Tour' from the Wharf or Transbay Terminal (Grey Line or Blue/Gold Fleet). That's easy because they'll do all the time management for you, but provides less freedom to explore. If you're a tourist and don't want to drive, this is probably the safest way to go.
However, should you want to go on your own using public transit, it's a little more complicated:
- Take a Vallejo bound Baylink Ferry/Bus from the Downtown Ferry Building at the foot of Market St.
Ferry/Bus takes 1 hour from San Francisco to Vallejo.
- Once you're in Vallejo, transfer to a Number 10 VINE Bus northbound (bus stops in front of the dock). It will go through the Napa Valley all the way to Calistoga. There's one bus per hour from 6 AM to 9 PM. The Bus leaving Vallejo gets to Napa in about 1 hour, to Calistoga in 2 hours.
That's 3 hours one-way if you go through the entire valley (not counting stops for shopping, tasting, eating).