Combine the butter, lemon juice, garlic, salt, pepper, and basil in a medium saucepan and cook over low heat, stirring until the butter melts. Cover and keep warm over low heat.
Heat the oil in a large skillet over medium heat. Season the fish with salt and pepper, to taste. Cook the fish for 3 minutes; then turn and cook until just opaque, about 3 to 4 minutes more. Transfer the fillets to individual plates.
Spoon the warm basil butter over the fish and serve.