Combine the maple syrup and freshly squeezed lemon juice in a large pitcher. Stir in the water or sparkling water.
Add the blackberries and basil to a food processor. Puree until the berries are completely broken down.
Transfer the blackberry-basil puree to a fine mesh sieve placed over a bowl or over the pitcher. Using a large spoon, continuously press the puree through the sieve so the juice goes into the bowl/pitcher and all the fruit pulp and seeds are left in the sieve. Reserve the pulp and seeds for another use, such as smoothies or jam.
