July and August are the busiest months, when Greek families, European backpackers, and ferry-hoppers all descend at once. The beaches are crowded by Cycladic standards, though by the standards of, say, Mykonos they're still bearable. The meltemi — the strong northerly wind that sweeps the Aegean every summer — blows hard in August, which is unpleasant in town but ideal if you're at Mikri Vigla to windsurf. Expect it.
June and September are better choices for most visitors. The weather is reliably warm and sunny, the ferries run frequently, and the island hasn't quite emptied of tourists — so restaurants and bars stay open. Sea temperatures are at their peak in September, and the light turns golden in a way that July's relentless bleach doesn't quite manage.
May and October are for people who want the island to themselves. Some beach tavernas and tourist-facing businesses close, but the villages, the archaeological sites, and the mountain roads are entirely accessible. The climate is mild and often lovely. Bring layers for evenings.
Winter sees a quiet, fully local Naxos. The population is around 18,000, and they get on with island life — fishing, farming, arguing about local politics — in a way that summer obscures completely. A few restaurants stay open year-round. It's worth knowing that the ferry service thins considerably between November and March, and bad weather can cut the island off for days at a stretch.