I am pretty sure you can chalk this one up to "family legend."
Per Wikipedia: http://en.wikipedia.org/wiki/San_Francisco_Bay

In the 1860s and continuing into the early 20th century, miners dumped staggering quantities of mud and gravel from hydraulic mining operations into the upper Sacramento and San Joaquin Rivers. GK Gilbert's estimates of debris total more than eight times the amount of rock and dirt moved during construction of the Panama Canal. This material flowed down the rivers, progressively eroding into finer and finer sediment, until it reached the bay system. Here some of it settled, eventually filling in Suisun Bay, San Pablo Bay, and San Francisco Bay, in decreasing order of severity.
By the end of the 19th century, these "slickens" had filled in much of the shallow bay flats, raising the entire bay profile. New marshes were created in some areas."

"Large ships transiting the bay must follow deep underwater channels that are maintained by frequent dredging as the average depth of the Bay is only as deep as a swimming pool--approximately 12 to 15 ft (3.7 to 4.6 m). Between Hayward and San Mateo to San Jose it is 12 to 36 in (300 to 910 mm). The deepest part of the bay is under and out of the Golden Gate Bridge, at 372 ft (113 m)"

In all the stories I heard first hand about the Great Fire and Earthquake, sunken ships were not a part of them. Even my cousins who descend from commercial fisherman only said the boats were taken to Sausalito or the East Bay.

A "ship" would not likely sink in about 12' of water. However -- look to reports of sunken ships off the coast of San Francisco. Most family legends have some fragment of a truth... whether it actually tracks to your family or not may be a different story!