Place the rack in the top third of the oven and preheat the broiler.
In a small saucepan, combine pineapple juice, butter, lime juice, lime zest, Tabasco, ginger, garlic, basil, salt and pepper. Heat and simmer for 4 minutes. Stir in rum and heat for 2 minutes.
Remove 2 tablespoons of sauce to a small bowl. Blend in cornstarch and add back to the sauce. Cook over low heat, stirring constantly, until thickened enough to spread.
Place fish on lightly greased broiler pan, skin side down. Brush with sauce mixture. Cook about 8 to 10 minutes or until done. Brush again with glaze about halfway through cooking time. Keep remaining sauce hot.
Remove to a serving dish. Spoon remaining sauce over fish.
Contributed by Joyce Taylor