Stir together 1 cup water, vinegar, sugar, and salt in a medium saucepan. Bring to a boil over high, stirring until sugar dissolves. Remove from heat; cool 10 minutes. Place basil, shallots, crushed red pepper, and green tomatoes in a 32-ounce canning jar. Pour vinegar mixture over green tomato mixture in jar. Cover with a tight-fitting lid; chill 24 hours. Store in refrigerator up to 2 months.