The vehicle had 11 people on board when it ran off the road near the popular Marina Grande area of the island.
Capri mayor Marino Lembo told Italian state TV that the bus driver died shortly after the crash.
Four passengers were seriously injured, including a child who was flown by helicopter from the island to a hospital in Naples, Italian media reported.
Several other passengers sustained lesser injuries, Italian media said.
The state TV report said the bus plunged some five to six metres (17-20ft) off the road.
The bus appeared to have smashed against some structures when it landed away from where people were sunbathing, but it was unclear whether they were changing cabins.
Italian media said some beachgoers instinctively dove into the sea for cover when they heard the loud crash.
Capri’s mayor said the cause of the crash wasn’t immediately known, daily newspaper La Repubblica reported.
