In a small saucepan, stir together the water, sugar and berries over medium heat until the sugar has dissolved. Bring the mixture to a boil. Muddle the berries to break the skin and bring heat down to a simmer. Cook for an additional 10 minutes then remove from heat.
Crush a handful of fresh basil leaves and place in hot simple syrup. Allow to steep for about 10 minutes.
Strain the syrup into a bowl and chill.
To Make the Cocktail
Add ice to a cocktail shaker. Pour in the 3.5 oz of the syrup, 1.75 oz of the key lime juice and 1.75 oz of vodka. Shake and strain into a cocktail glass. Garnish with a blackberry.