Heat oil in large saucepan over medium heat. Add white and pale green parts of green onions, 1 tablespoon cilantro, and garlic; saute until tender, about 2 minutes. Add curry paste; cook until fragrant, about 1 minute. Add 1 1/4 cups water, coconut milk, chiles, lime leaves, and fish sauce. Bring to simmer. Add carrot; cover and cook until carrot is just tender, about 5 minutes. Layer bok choy, shrimp, scallops, and mussels in pan. Cover and simmer until mussels open and seafood and bok choy are cooked (discard mussels that do not open), about 5 minutes. Stir in dark green parts of green onions, 2 tablespoons cilantro, and basil.

Divide rice among 4 shallow bowls. Ladle curry over rice and serve.