Bali has two seasons: the dry season (May–September) and the wet season (November–March). Denpasar specifically is worth planning a visit around one event above all others: the Bali Arts Festival, held at the Taman Werdhi Budaya Art Centre every June and July. The month-long programme of traditional music, dance, and craft from every corner of the island is the single best cultural event in Bali and something that most visitors miss entirely by going during the peak beach season.
Outside the Arts Festival, the dry season is more comfortable — lower humidity, clearer skies — but the city functions year-round. The Hindu religious calendar produces frequent temple ceremonies and processions throughout the year; the dates of major festivals (Galungan, Kuningan, Nyepi) shift each year according to the Balinese calendar. Nyepi, the Balinese New Year's Day of Silence, is the most dramatic: the entire island falls silent for 24 hours, no lights, no movement, no noise — including at the airport, which closes.
The beach resort towns to the south and west (Kuta, Seminyak, Canggu) are busiest July–August and over the Christmas/New Year period. If you prefer the city quieter and cheaper, May–June is an excellent window: dry, before the peak-season rush, and the Arts Festival is starting.