Paul Harris remembers driving to the New Jersey shore in a Ford Model A to go fishing with his father.

“Back in the 1940s, we’d go to the old Phipps estate for the weekend and fish for kingfish and croakers. Then we’d drive back home to Philadelphia, where the mothers and grandmothers were all waiting for the fish,” says Mr. Harris, 75, who still fishes that 10-mile stretch of shoreline, now known as Island Beach State Park.

Mr....