Aasiaat is one of the better whale-watching bases in Disko Bay. In summer, humpback, minke, and other whales move through the waters around the islands, sometimes close enough that the first sign is the sound of a blow across still water.
Small-boat trips give the best chances, but you do not need a polished safari setup to enjoy it. Keep binoculars handy on coastal walks and ferry approaches, and give whales plenty of space if you are out by kayak or private boat.