Upvote:3
An airport makes its money from fees charged to user airlines. These fees include (but are not limited to)
Whether there is a specific charge for a dedicated area in the check-in hall, or a per-hour charge for the use of check-in desks, or the fees are rolled up into some other charge, will be the subject of the contract between the airline and the airport, and may differ from airline to airline and airport to airport. The exact charges levied may also be commercially sensitive.
The provision of check-in desks has to be paid for somehow, so yes, one way or another, the airlines pay.