Place the shrimp in a medium bowl. In a blender or food processor, combine basil, coconut milk, garlic, lime juice, ginger, soy sauce, fish sauce, and brown sugar. Blend until smooth. Pour over the shrimp and toss to coat. Cover with plastic wrap and marinate for at least 1 hour (or up to 4 hours). Preheat the grill to medium-high heat. Thread the marinated shrimp onto the grill, leaving as much marinade on the shrimp as possible. Grill for 2-3 minutes per side, or until pink, cooked through, and slightly charred.