Think of what consequences there may be if this gets into the press. While the person was playing poker, you state that it was front desk staff playing poker in the lobby with a guest.
Firstly, be completely sure they were in a situation that could be called a dereliction of duty. If their duty is to welcome guests and just "be there," they could easily claim they were.
Secondly, as a hotel, you should be welcoming to your guests (or should I say paying customers). This person is the face of the hotel at night, welcoming back drunks and receiving complaints from customers with issues. The person they were playing cards with probably thinks your hotel is great, and the staff are fantastic, and may have shared this with their friends. Free publicity that your hotel could not dream of buying!
If you take draconian action, this person could claim wrongful dismissal or constructive dismissal. The press would seize this as a story about a big bully hotel victimizing an employee who was trying to be nice to a guest. Watch your bookings drop if this were to happen.
As someone suggested, provide e-learning and other activities that can be done so that the person is not just sitting around with little to do; and perhaps have a quiet word about how some guests may perceive this as inappropriate and that a polite "sorry" should be offered to any guest wanting to join the game. Involving HR could lead to an upset, disgruntled employee, especially if they believe they were doing no harm.