How to clean a cork bottle stopper?
From  hunker , here is info and instructions for sterilizing corks:

Corks are used to cap wine and other types of bottles. Before corks are used to seal bottles, they are cleaned and sterilized to prevent any contamination. It is relatively easy to sterilize corks for reuse by steaming or boiling them for an extended period of time. Once the corks have cooled and dried, they can be safely used to store homemade wine, handmade vinegar or flavored olive oil.
Step 1
Add sufficient water to the saucepan so that it won't boil dry during the sterilizing period. Add the corks to the pan. If you are using a steamer or steamer basket, fill the pan with water to the bottom of the basket.
Step 2
Add the corks, cover the pan and boil or steam for 90 minutes. While the corks are boiling, sterilize the tongs, jar and lid or container by immersing in boiling water while the corks are being heated. Once the implements have been in the boiling water for five minutes, drain and place them on a clean dish cloth. Check the corks to make sure they have enough water.
Step 3
Drain the water off the corks once the 90 minutes are up. After the corks have cooled and dried completely, use the tongs to place the corks in the container or jar and cover and use as needed.

Also, for just cleaning and not sterilizing, this  article from hunker  will take you through the steps of cleaning cork stoppers.