October to March is the best time to visit. The monsoon retreats by late September and the coast settles into clear skies, calm seas, and comfortable temperatures in the mid-twenties. This is when the beaches are at their best, boat crossings to Devbagh and Kurumgad are reliable, and the forest roads to Dandeli are passable without difficulty.
April and May are hot — temperatures push into the mid-thirties and the humidity builds before the rains arrive. Visiting is still possible but less comfortable, and the Dandeli forest trails can be dusty and difficult.
June to September is monsoon season. Karwar receives heavy rainfall — the Western Ghats intercept the Arabian Sea moisture and the coast gets the full effect. Boat services to the islands are suspended, the sea is rough and grey, and the beaches are deserted. The river and the jungle turn intensely green, and if you like the drama of an Indian monsoon this is one of the better places to experience it, but practical travel is much harder. The Kali River can flood.
The Shivratri festival at Kurumgad Island draws large numbers of pilgrims — typically February or March — and boats to the island are crowded in the days around the festival.