Stir together blueberries, halved strawberries*, basil leaves, and sugar in a medium bowl; crush berries and leaves with a wooden spoon. Chill 30 minutes (to release juice from berries). Stir 1 cup chilled sparkling wine into fruit mixture, and pour through a fine wire-mesh strainer into a pitcher, discarding solids. Add remaining wine to pitcher. Garnish, if desired. Serve immediately, or chill up to 24 hours.
*Frozen, thawed strawberries and frozen blueberries may be substituted.