Peel one of the cucumbers, cut it in half, and then cut each half in half lengthwise. Scoop seeds out with a spoon, discard, and dice cucumber.
Put cucumber pieces in a food processor or blender and puree until smooth.
Put cucumber puree through a fine mesh sieve over a bowl and push with a wooden spoon or spatula, extracting as much liquid as you can from the cucumber puree. You should be able to get about ¾ cup.
Pour cucumber juice into a large pitcher with ¾ cup lemon juice, water, and granulated sugar. Stir in vodka if using. Slice remaining half cucumber and add cucumber slices, lemon slices, and fresh basil to pitcher.
Let mixture chill until ready to serve. Pour cocktail into ice-filled glasses, making sure to get some sliced cucumbers and lemons and basil in each glass.