You can dive in Indonesia at any seasons and there are liveaboards operating in various parts of the country throughout the year. With such a vast and geographically diverse country, it is impossible to give general guidelines. What is possible is to state that Indonesia, being at the heart of the Coral Triangle, boasts unbelievable biodiversity and across the range of destinations within the country, an amazing array of choice. Where in the country you choose to go and at what time of year depends on many factors, not least what type of scuba diving and creature encounters you are seeking.
Komodo has year round diving with some people considering April to November to have the very best conditions, and August in particular being the best month for Mola Mola. Outside of this time plankton blooms may reduce visibility a little but this increases the chances of manta ray encounters which are most frequent between December and February.
Raja Ampat and the West Papua Province have some liveaboard charter boats operating all year although many restrict their season to October to April when there is less chance of rain and choppy seas. Mantas are more frequently sighted during these months. Sea temperatures do not vary much during the year but the frequency of rain, in this amazing (but relatively wet) destination is a seasonal factor.
North Sulawesi is another destination that can be dived all year round. Manado and the Bunaken National Park enjoy the best conditions from March to October with July and August being the busiest months. July and August are also the best months for critter spotting in Lembeh, although this too can be amazing during any given month. November to January are the months where you can be least certain of great conditions throughout North Sulawesi.
Bali is best dived between April and November, with sightings of Mola Mola, sharks and other pelagics tending to be most frequent between June and September, and manta rays between April and June. Outside of this time is officially rainy season, although conditions are often excellent during then too, with visibility a little compromised off the north and west of Bali where less diving is done.