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I'm not sure if this is feasible and certainly it would be more of a process than 'building some kind of dam'. A few reasons:

- The bridge spans 2700m and the channel is 350+ feet deep in some spots. For scale, the Three Gorges Dam is 2300m long

- Sea levels will rise so incrementally that a massive public works project might seem unnecessary or hard to get support for

- Shipping traffic needs to be able to pass through to get to the port of Oakland, Richmond, Stockton, etc...

Where would the water from all the rivers that empty into the bay go?

The worlds only 1m high hydroelectric dam.

Alas, since you want to keep the water level lower inside, that means you'd have to pump the incoming flow up to get it into the ocean. Rather than generating electricity, it would consume vast quantities of it.

The tide moves 6ft+ from high to low. As long as the flow rate is less than 3ft/day then you can just wait for low tide.

This map is simply based on sea level and totally ignores sea force. If the sea were 10' higher, it would probably erode most of San Francisco within 50 years.