Prepare dip. Place mayonnaise, cheese, basil and garlic in food processor bowl; process until smooth. Spoon dip into small bowl; cover and refrigerate until ready to serve.
Heat oven to 400F. Wash and trim Brussels sprouts. Cut an "X" in stem end of each sprout. Place Brussels sprouts in medium bowl. Add remaining ingredients; toss to coat. Spoon seasoned sprouts onto baking sheet.
Bake 25 to 40 minutes, stirring frequently, until sprouts are tender and golden brown. Serve warm or chilled with Creamy Parmesan Basil Dip.