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