Prepare simple syrup by bringing water and sugar to a boil; remove from heat and stir in fresh basil leaves and blackberries.
Use a potato masher or fork to mash blackberries and basil.
Let stand until mixture is cooled, then use a fine mesh strainer to strain.
In a medium pitcher, combine lemonade, vodka, water, and strained simple syrup.
Garnish with blackberries, lemon slices, and fresh basil.