The Marshall Islands has two seasons, and the difference matters more for comfort than for access.
The dry season runs from December through April. Trade winds keep temperatures down, rainfall is low, ocean visibility is at its best, and seas are generally calm. This is the best time to visit for diving, snorkeling, and outer island travel. March is particularly reliable.
The wet season runs from May through November, with the heaviest rainfall from July to October. Humidity climbs, showers come daily, and occasional tropical storms pass through. Diving is still possible — wreck dives around Kwajalein are excellent in calmer wet-season conditions — but outer island transport becomes less predictable.
Water temperatures stay between 27°C and 29°C year-round, so there is no bad season for swimming. The main consideration for most travelers is whether you want to guarantee good conditions for diving and boat travel, in which case December to April is the answer.
Note that the outer islands can become nearly inaccessible during extended bad weather even in the dry season. If you are planning to visit Bikini Atoll — which requires a live-aboard dive boat or a special permit — plan well ahead and expect the unexpected.