I don't know of any other way to Staithes, at least, no other way I'd be interested in short of boaating.
Make Whitby your base and drive out on the A174 to Staithes... Hell, if you get tanked or fall in love with Staithes - splurge and stay the night in one of the billion cottages in the old town.
The old city of Staithes has to be done by foot. Period. Walk down from the parking lot above the old city.