In a small saucepan, add the blueberries and basil. Cook, mashing the blueberries as you go until it cooks down completely. Strain through a fine mesh strainer into a container. Chill in the refrigerator until cold. This can be up up to three days in advance.
Fill your glass with ice.
Pour the blueberry basil syrup and gin in a cocktail shaker with a few ice cubes. Shake to combine and strain into your glass.