Remember, there is a special corkscrew made for corks that are dried out or broken. The one commonly used is the “Ah-So” corkscrew.
It will pay you to know if your restaurant has one so if you sense a cork is breaking when you use the regular pull-tap corkscrew, you can go get it and rescue that bottle from having little bits of cork in the wine.
Most guests will recognize that it is not your fault that a cork crumbles and if you are able to “rescue” that wine from getting pieces of cork in it, they will appreciate your wine service that much more.
The Ah-So cork remover can also be used routinely to open older bottles (>10 y. o.) so that you don’t get into the problem of crumbling and old cork.