For almost all delicate green herbs, the best thing to do is
1 Hold them by the bottom of the bunch and gently swirl in plenty of cool water to remove all the dirt
2 Gently shake to remove excess water
3 Wrap loosely in a damp paper towel
4 Place in a plastic bag or in an airtight container and store in the crisper drawer of your refrigerator
3 days later:
This will keep most herbs fresh for many days.