Monterey Regional Airport: The closest sensible airport. Rent a car at the airport and drive via Carmel: roughly 45–60 minutes to Big Sur Village in normal conditions. There is no airport transfer that makes Big Sur workable without a car; budget for a rental plus high-priced Highway 1 fuel.
San Francisco International Airport: Rent a car and take US-101 south, then CA-156 west to Highway 1 at Castroville; allow 2–3 hours to the northern Big Sur boundary in clear traffic, longer on Friday afternoons and weekends. This is the practical Bay Area approach, not a shuttle route.
Los Angeles International Airport: For the southern approach, rent a car, take US-101 north to San Luis Obispo, then Highway 1 via Morro Bay/Hearst Castle. It is about 4–5 hours to the southern boundary in good conditions, and significantly longer to Big Sur Village. Do not commit to this route without checking Highway 1 status first.