Sikinos is at its best in May–June and September–October, when the weather is warm, the meltemi (the Aegean summer wind) has not yet hit its peak, and the island is quiet enough that you can hear yourself think. July and August bring more visitors than Sikinos is used to — still far fewer than the big Cyclades — but the wind can be fierce and the heat intense.
The ferry connection from Piraeus operates year-round but becomes less frequent in winter. Off-season, some tavernas and accommodation close, but the Hora always has life. Spring is particularly lovely: the hillsides are covered in wildflowers, the air is cool, and the walking paths are at their best.
Avoid ANZAC Day weekend if you are not Australian or New Zealand and you happen to be on a Gallipoli tour; Sikinos itself has no particular seasonal events beyond the monastery festival on the first Friday after Easter.