Allow soup to stand 30 minutes at room temperature before serving. The flavor will come through better if you take the chill off. Slice your basil and chop the avocado just before serving so it doesn't begin to brown.
Combine broth and cornstarch, stirring with a whisk.
Melt butter in a large saucepan over medium-high heat. Add onion; sauté 3 minutes. Add garlic; sauté 30 seconds. Stir in broth mixture, corn, salt, and pepper; bring to a simmer. Cook 10 minutes, stirring frequently (do not boil). Place half of corn mixture in a blender, and process until smooth. Pour pureed corn mixture into a large bowl. Repeat procedure with remaining corn mixture. Strain corn mixture through a sieve into a large bowl; discard solids. Stir in milk and half-and-half; chill thoroughly. Let stand 30 minutes at room temperature; stir well.
Ladle 2/3 cup soup into each of 8 bowls; top each serving with 3 tablespoons crab, about 2 teaspoons avocado, and about 1 teaspoon basil.