We book a lot of Mykonos. The single most useful thing we tell people is also the thing they least want to hear, which is that August is the worst month to go and September is close to perfect.
What actually changes
The sea in September is at its warmest, because it has had all summer to heat up. The bars in Chora are still fully open. The beach clubs are still running. What falls away is the volume of people and the price of everything. August on Mykonos means beaches filling by mid morning, restaurants with waits, and sunbed minimum spends at the fashionable clubs that run into several hundred euros before you order.
June is the other sweet spot for the same reasons in reverse. The island has woken up, the crowd is thinner, and the light in the evenings is beautiful. The sea is a touch cooler than September, which almost nobody minds.
The gay scene does not shut down
The worry we hear is that going outside August means an empty island. It does not. The gay bars in Chora, Jackie O', Porta, Lola and Cosma, run right through September, Pierro's is still going until sunrise, and Elia is still the gay beach. What you lose is the queue, not the scene.
The one reason to go in August
XLSIOR. If a festival of 30,000 people is what you want, August is the only time it exists, and no amount of sensible advice about crowds is relevant. Go, enjoy it, book everything a long way ahead.
But if what you actually want is Mykonos itself, the town, the water, the food, a bar crawl that ends when you decide rather than when the crush pushes you out, then you are paying a heavy premium in August for a worse version of the island.
May and October
Lovely and quiet, with a caveat we would rather say out loud. Some beach clubs are not open yet in May, and by late October bars run shorter hours and the island starts closing around you. If the nightlife is a main reason for the trip, stay inside June to September. If it is a bonus, the shoulder months are excellent value and genuinely romantic.
More on this in the Gay Mykonos guide.