1. Make basil syrup: simmer water and sugar until dissolved, remove from heat, add basil, steep 20 minutes, strain, and chill.
2. Blend blackberries and strain to remove seeds. Combine gin, lemon juice, blackberry purée, chilled basil syrup, and cold water in a pitcher; stir and chill.

3. To serve, fill tall glasses with ice and pour 6–7 oz per drink. Garnish with a basil sprig and a couple of blackberries.