Start by cutting your stalks of lemongrass into 1-inch sections. Bruise them with a meat mallet/rolling pin/mortar and pestle. Set them aside for a moment.
Add the sugar and water to a small saucepan over medium heat. Stir occasionally until the sugar is entirely dissolved. Allow it to come to a boil, then turn off the heat. Add the lemongrass and allow it to steep for 2 hours.
Strain the syrup, pressing down on the lemongrass to extract all of the flavor. This simple syrup will keep in the fridge for about 4 weeks.
To make the cocktail
Add the slice of lime and basil leaves to the bottom of a glass. Muddle them together.
Add the rum, simple syrup, and ice. Top off with some Coconut Lacroix or club soda.