Stir together sugar, 1 cup water, cucumber, cardamom, and basil in a small saucepan. Bring to a low boil over medium, and cook, stirring occasionally, until sugar is dissolved and syrup is slightly thickened, about 8 minutes. Cool 20 minutes. Pour mixture through a strainer into a bowl, discarding solids. Stir in salt. Syrup can be stored in an airtight container in the refrigerator for up to 1 week.