Simply close them
It is true that closing credit card can hurt your credit. It makes your credit history younger which is bad for your score (Length of credit history counts for 15% of your fico).
Closing cards also lowers you available credit and raises your balance-to-limit ration which is also bad for your score (Utilization counts for 30% of your fico). However, if you carry no balances then it doesn’t matter anyway.
By the way, some of these cards may have already been closed due to lack of activity. You may want to check your credit report to verify their status. Closed accounts in good standing remain on your report for at least 10 years. They just don’t count as much in your score as open, active accounts. So, since you are not using these cards, they really don’t do much to help your credit anyway.
Having 13 unused cards in you drawer is also asking for trouble with fraud and ID theft. What I would do is close some of these cards. I would keep the oldest open (that way you don’t loose credit history length) if it has no annual fee.
Keep your oldest card only if it’s older by more than 4 years from the other cards. If the card is less than 4 years older than your other cards it makes no difference.
Closing these 13 accounts may drop your score 50 points, but even if it does, you score will rebound within a year.
Close the accounts via written letter and request written confirmation that the account is closed with zero balance. Keep that confirmation forever.
Please see How to Close Credit Card Accounts and sample letter for closing a credit card account for more information.