Kennebunkport is principally a summer destination. The peak season runs from late June through August, when the beaches are warm enough to swim in (just), the lobster boats are running, and every restaurant and shop is open. Summer weekends are crowded; weekdays are more manageable. Prices for accommodation peak in July and August.
September is arguably the best month: the water is as warm as it gets (still cold by most standards — perhaps 17°C on a good day), the crowds thin out dramatically after Labor Day, and the foliage starts turning gold in the surrounding countryside by mid-month. The light in September on the Maine coast is exceptional.
October brings the full fall colour and excellent hiking on the nearby trails, but many businesses close for the season from mid-October. Check before you go.
Winter is quiet and cold; most visitor facilities are closed, though a handful of year-round restaurants and shops remain open. The coast in winter has a stark, beautiful quality all its own.