Cook rice per package directions, about 20 minutes. Set aside and keep warm.
In a bowl, whisk coconut milk, 2 tbsp of the curry paste, 2 tbsp of the lime juice and the ginger. Place shrimp in a resealable plastic bag, add coconut milk mixture and marinate 15 minutes. Heat grill to medium-high.
In a large bowl, whisk remaining 1 tsp curry paste, remaining 1 tbsp lime juice and the oil. Toss pepper pieces in bowl to coat. Season with 1/4 tsp of the salt. Grill pepper pieces 10 minutes, turning frequently. Remove to a cutting board.
Thread shrimp onto skewers, reserving marinade. Grill shrimp skewers 6 minutes, turning once, until cooked through. Season with remaining 1/4 tsp salt and the pepper. Meanwhile, bring 1 cup of the reserved marinade to a boil (discard remaining marinade). Boil 3 minutes. Slice peppers into thin strips.
Spoon rice onto a platter. Top with pepper strips, shrimp and basil. Drizzle a little sauce over platter and serve remaining sauce alongside.