Local's secret
As long as we are discussing the best time to do this ride....Here's the answer if you are very flexible on dates and advance notice....
Wait for a Big winter storm. The road through Big Sur often closes for a few days to vehicle traffic. If you check with CalTrans you can find out the status of repairs and reopening of the road.... You can start from Monterey (generally a day or two before the road reopens as the slides tend to be toward the southern end between Ragged Point and Gorda) and be on the road with almost no cars...you have to talk your way across the repair spots but usually you will be allowed through if they are at the end of the repairs but cars will not be allowed...
Other than that.... Fall is best for no rain, Winter is great if you have a good long range forecast, spring is beautiful and green but can be wet also, summer is foggy and crowded
I agree with VLYGRL about Kirk Creek Campground but you might also want to consider the campground just above it at Jade Creek called Limekiln... It used to be a private campground and then was turned into a State Park....SHOWERS available there...