Wine corks dry out for a number of different reason, one of the most common being improper wine storage. If the corks are dry, either from being stored in the wrong position or other issues, there are different methods you can attempt to re-moisten them. By fixing the dried out wine corks you can avoid pushing the cork into the wine or ruining the wine when air is allowed in through gaps between cork and bottle.
Set the bottle of wine on its side to allow the wine to soak the inside of the cork for 24 hours. The cork will absorb a small amount of the wine and bring moisture back into the cork.
Boil a small amount of water on the stove and soak a washcloth in the water. Put an oven mitt on one hand and pick the washcloth out of the water using tongs with the other hand. Hold the washcloth in the oven mitt and place it over the cork. Slowly try to twist the cork out of the bottle; the boiling water should moisten it and loosen it up.
Remove the cork from the bottle using an ah-so wine opener. An ah-so opener has two narrow prongs that slide between the bottle and the cork, you should be able to lift out the cork by rocking the cork back and forth, even if the cork is extremely dry, according to online resource Anything Wine.