My company recently performed a complete license audit and found that a couple of our Exchange Server Enterprise Edition systems are licensed only for the Standard Edition. Is there any way to downgrade the Exchange installation on those servers?

Not really. You can perform an in-place upgrade from Standard Edition to Enterprise Edition, but the reverse isn't possible. Assuming that upgrading your licenses isn't an option, you'll need to remove and reinstall Exchange. Make sure you have a good backup of your database before you begin.

Be aware that Exchange Standard Edition supports only one storage group (SG) and requires its lone mailbox database to be smaller than 16GB. If your existing database is bigger than 16GB, Standard Edition won't be able to mount the database; you'll need to shrink it before downgrading. First, decide how you want to divide the mailboxes on the existing server. Second, move the excess mailboxes to another server. Third, use Mailbox Manager to clean out old messages on the mailboxes that will remain on the original server (or on all the mailboxes). Finally, perform an offline defragmentation on the EDB file to shrink the amount of disk space that it uses.