Tarawa sits one degree north of the equator, so the climate is hot and humid all year round — daytime highs in the low thirties Celsius, nights barely cooler, and the sea always warm. There is no real "high season" in the tourist sense; you go when you can get a flight.
The drier months run roughly May to November, with steadier trade winds and slightly less rain. December to April is the wet season — heavier downpours, more humidity, occasional periods of very high tides ("king tides") that wash across parts of the road. El Niño years can shift this pattern significantly; check current conditions before you book.
If you want to time a visit around an event, Independence Day on 12 July is the largest national celebration of the year and brings dance, sport, and a festive atmosphere to South Tarawa for several days.