In a large bowl or kettle combine 1 quart of the cold water and 2 tablespoons of the salt; add mussels. Soak the mussels for 15 minutes; drain and rinse mussels. Discard soaking water. Repeat the soaking, draining, and rinsing process two more times.
In a 3-quart saucepan cook sweet pepper and garlic in hot margarine or butter just until crisp-tender. Stir in tomato, wine, and basil. Add mussels. Bring mixture to boiling; reduce heat. Cover and simmer for 3 to 4 minutes or just until mussels open. Remove from heat. Using a slotted spoon, remove mussels. (Discard any mussels that do not open.) Cover mussels and keep warm.
If necessary, simmer the sauce, uncovered, until it has been reduced to 1 cup. Pour the sauce over the mussels. Garnish with additional fresh basil and serve with French bread, if desired. Makes 2 main-dish servings.