Anamur is warm enough for swimming from May through October, with sea temperatures peaking in August. The sheltered position between the Taurus Mountains and the coast gives it a more temperate winter than the broader Mediterranean norm — frosts are rare — but most facilities outside the town centre close between November and April.
The shoulder months of May, June, and September are the best time to visit. Temperatures are comfortable for walking archaeological sites (Anemurium and Mamure Castle both involve significant time in exposed sun), the sea is warm, and the town has not filled with the Turkish domestic tourists who arrive in July and August. The beach stays busy with local families through August, which is loud but convivial; the ancient sites get more crowded but never oppressively so.
July and August bring the banana harvest — if you are interested in seeing the fruit collected and sold, the market days at the bazaar in those months are worth timing your visit around. Winter travel is possible but pointless unless you are specifically here for the archaeology, in which case the empty sites and cool temperatures are actually ideal.