Bring water, sugar, and ginger to a slow boil until the sugar has dissolved.
Turn off the heat and let the ginger steep for at least 30 minutes. The longer you let it steep, the more pronounced the ginger flavor will be (I generally leave mine for several hours). Once it's to your liking, strain the syrup to remove the ginger and chill in the refrigerator.
To make the lemonade, combine 1 part lemon juice (¼ cup) with 2 parts ginger syrup (½ cup) and 3 parts cold water (¾ cup). Pour into glasses and serve with lemon slices and a sprig of lemon basil.