Your grapes may have black rot, a fungal disease. Black rot cannot be cured once the grapes are infected, but the page linked to below details a number of integrated pest management strategies for this disease. Minimize black rot problems by removing infected fruit, cleaning up mummies, and pruning out leaves that surround fruit clusters.You will find more information from Missouri Botanical Garden here.

Grapes are susceptible to a number of other diseases. Penn State Extension describes their symptoms and management strategies, in the event that you look into black spot and determine that it is not what's affecting your plants.

Courtesy of NYBG Plant Information Service