Trim ends of zucchini. Using a sharp vegetable peeler, slice zucchini lengthwise into wide, flat ribbons. Discard first and last slices, and the seedy portions in the middle. (You will need 32 ribbons.) Set ribbons aside.
Carefully clean crabmeat, removing any shell or cartilage pieces. Drain crabmeat well in a colander, pressing with the back of a spoon to remove most of the liquid. Pat dry with paper towels. In a small bowl combine crabmeat, mayonnaise, wasabi paste, and salt. Seed and peel the avocado; cut into thin strips.
On a clean work surface, place one zucchini ribbon on top of another. For each roll-up, place 1 slightly rounded teaspoon of crab mixture at one end of a doubled zucchini ribbon. Top with avocado strips, a few shreds of carrot, and a basil leaf. Roll up; secure with toothpicks. If desired, cover and chill up to 30 minutes. Makes 16 appetizers.