Clean mushrooms with a damp cloth and remove their stems. Place in a glass dish, stemless (gill) side up.
To prepare the marinade, in a small bowl whisk together the vinegar, pineapple juice and ginger. Drizzle the marinade over the mushrooms. Cover and let marinate in the refrigerator for about 1 hour, turning mushrooms once.
Prepare a hot fire in a charcoal grill or heat gas grill or broiler. Away from the heat source, lightly coat the grill rack or broiler pan with cooking spray. Position the cooking rack 4 to 6 inches form the heat source.
Grill or broil the mushrooms on medium heat, turning often, until tender, about 5 minutes on each side. Baste with marinade to keep from drying out. Using tongs, transfer the mushrooms to a serving platter. Garnish with basil and serve immediately.