There is hardly anything you should need to do to care for dissolvable stitches in the earlobe. They are designed to go away after a period of time. If they do not, then the surgeon can remove them. Sometimes they can go away faster if you have been instructed to put ointment on top of them.
Different surgeons have different protocols that work best for them. Personally I have patients apply antibiotic ointment twice daily until they are seen a week later for suture removal. Showering is permissible to keep the sites clean. Contact your surgeon to best understand what post operative instructions they would like you to follow.