Capri is limestone cliffs, not beaches. There are precisely 2 places to swim from sand/pebbles, and a handful of rocky bathing platforms (stabilimenti). The Blue Grotto is for LOOKING, not swimming (officially). The real swimming: rent a gozzo boat (120-250/day) and jump off wherever the water calls.
THE swimming spot. Pebble beach with Faraglioni views. Stabilimento da Gioia or Lo Scoglio. Deep water immediately. 30-60/day for a sunbed.
Pebble beach near ruins of Tiberius' palace. Boat from Marina Grande (5, 5 min) or 20 min walk. Restaurant on the beach. Less crowded than Marina Piccola.
A bathing platform at the base of the Faraglioni. Lunch at the restaurant (60-100/person). The most dramatic swimming platform in Italy. BOOK AHEAD.
A pool + bathing platform at the Punta Carena lighthouse. Sunset-facing. The locals' spot.
Rent a gozzo (small traditional boat, 120-250/day from Marina Grande). Swim at the White Grotto, the Green Grotto, the Faraglioni base, any cove you find. THIS is how Capri was meant to be experienced.