No new power infrastructure is needed...this really isn't that complicated
This isn't like turning on a light or the AC, where you need to have power available at that time. You can operate the desalination plants with off-peak energy, as well as buffer the variable demands (and supplies) during peak energy usage.
Given that this is a problem that has built up for years, and California is draining its aquifers even when they aren't experiencing a drought, you'd want to build the desalination plants so that they would be operated regularly (not continuously, we're using off-peak energy remember?)
Lewis' suggestion of adding 2% to California's energy usage to fix the problem over six years seems like a good initial target for the level of desalinization infrastructure you'd want to build. Enough to get "back to normal" in six years, and pumping the water into the aquifers to make up for those losses down the road. You can always add more plants down the road if the drought continues or worsens, or operate them less often if California starts setting yearly rainfall records.
OK, what about the environmental impact? California wants to reduce its CO2 impact, not permanently increase it by 2%! So how about wind or solar? The state I live in generates a quarter of its electricity from wind power. California's population is 13x larger, so if they added as much generation capacity as my state has, that would be their 2%. Though the desert may point to solar being a simpler alternative there - I merely pointed out the comparison with my state to forestall those who will claim "do you know how much energy California uses, there is no way renewable energy can produce that much".
The desalination plants wouldn't be directly powered by solar or wind, that would be added into the grid, and the desalination plants would act as buffers by using power that the rest of the grid doesn't need at the moment during peaks, and running on a more continuous basis during off-peak hours.
In this way renewable energy could solve California's water problems - and the water bills of California residents would pay for it. Using renewable energy may cost more today than just adding another gas fired plant or two, but getting new fossil fuel power plants built in California is not easy. Wind & solar is (at least comparatively speaking)