In a 4-quart Dutch oven or kettle combine apples, onion, and basil. Add vinegar and brown sugar. Heat over low heat.
Meanwhile, rinse blueberries, discarding any that are blemished; drain. Add to vinegar mixture in Dutch oven. Heat to boiling; reduce heat. Simmer, uncovered, for 20 to 30 minutes or until desired consistency.
Ladle hot mixture into hot, clean half-pint canning jars, leaving 1/2-inch headspace. Wipe the rims and adjust lids. Process filled jars in a boiling-water canner for 10 minutes. Remove the jars from the canner; cool on racks. Store in a cool, dark place up to 6 months. Makes 6 half-pints (84 one-tablespoon servings).