All of the above posters have offered suggestions that have merit, some with the forethought and perspicacity to understand that launching and vehicle storage fees may be an issue.
If I were advising someone coming to the Bay for exploration, especially from as far as Canada, I would suggest not staying in any one marina but moving from place to place.
Brisbane is an excellent choice for the south bay (inexpensive and good facilities). Foster City is worth a visit. I would recommend against the East Bay South of Alameda because of depth problems.
The gem is South Beach Harbor, especially if you are a good walker and/or bring a bicycle but it is maybe double the price of other places.
Berkley is certainly worth a visit and Richmond, located at the site of the old Kaiser shipyards, has historical interest. There is a great walking/biking path that runs from Richmond to Emeryville.
Depending on whether you have ornithological or fishing interests a trip into the rivers, Napa, Petaluma, San Joaquin and Sacramento might be enjoyable. In the cold months the bay will actually be warmer because to the east it can be foggy, grey and cold. In the warm months there is hot hot weather and fresh water swimming in the delta.