From Spring through Summer the miles of remote Wilderness State Park beach are closed to dogs (it's a breeding / nesting area for the rare Piping Plover). Come Fall---specifically September 1---all of that shoreline along Sturgeon Bay and the Straits of Mackinac becomes a gorgeous option for a dog outing.
There's a good chance that you'll have the beach to yourself, so you might be tempted to take your dog off leash. Keep in mind that Michigan State Park rules apply. Plus, you'll want to make sure to have a supply of poop bags with you (this is an ecologically sensitive area).
In addition to the beach areas, there are miles of great hiking trails throughout the 80,000 acre park. Be sure to check 'em out!