Marshrutka (minibus) no. 142 runs frequently between Batumi and Gonio, continuing south to the Turkish border at Sarpi. The journey from central Batumi takes about 25 minutes and costs a couple of lari. Get off at the fortress — the driver will know if you ask.
Taxis from Batumi cost considerably more but are still inexpensive. If you are driving down the coast to Turkey, Gonio is essentially on the main road; the fortress is signed from the highway.