For the smoothie:
In a food processor, combine the yoghurt, basil, lemon simple syrup and ice. Process until smooth and frothy. Pour into chilled glasses and serve immediately.
For the lemon simple syrup:
In a small saucepan, combine the water, sugar, and lemon zest over medium heat. Bring to a boil, reduce the heat and simmer for 5 minutes, stirring occasionally, until the sugar has dissolved.
Take the pan off the heat and allow the syrup to cool, about 20 minutes. Strain before using.