Mix Swerve and hot water together in a glass measuring cup, stirring to dissolve.
Combine lemon juice, cold water, vodka and Swerve mixture in a large pitcher. Stir well.
Tear basil leaves roughly, and place in the bottom of 4 tumbler glasses. Use a muddler or the back of a wooden spoon to crush basil leaves until they release some of their juices.
Add ice to glasses and divide lemonade mixture between them. Stir and enjoy.