The high season is mid-June through Labor Day weekend in early September: the beaches are full, the restaurants are booked out, and the ferry runs at full schedule. It is not Cape Cod-busy, but Lewes does fill up, and you should book accommodation and dinner in advance if you come in July or August.
The shoulder seasons — May, early June, September, October — are the best time to visit. The weather is reliably warm, the ocean is swimmable from mid-June through September, and you can get a table anywhere you want without booking. Cape Henlopen is at its best in early June and September: the dunes are quieter, the dolphins are still around, and the migrating birds are visible from the marshes.
Winter is quiet. Some restaurants close for a few weeks in January or February, and the ferry runs a reduced schedule, but the historic district is still walkable, and a winter beach walk on Cape Henlopen has its own appeal. The annual Punkin Chunkin festival (when it is held) and the Christmas events around the town clock are worth catching if you are in the area in late autumn.