Mykonos does not have a developed gay sauna scene, and this is worth being honest about before you arrive expecting one. The island's identity as a gay destination is built around the beach, the bars, and the outdoor social circuit. Saunas are not part of that picture.
Why this is the case
Mykonos is a small Greek island with a season that runs roughly June to September. The crowd is large in summer but the population outside that window is minimal. The kind of year-round infrastructure that supports a sauna scene does not exist here. The socializing happens at Super Paradise Beach during the day and in the bars and clubs of Chora at night. That is the format Mykonos is built for.
Where to go instead
If a gay sauna is specifically what you are looking for, Athens is the answer. The Athens sauna scene is well-developed, year-round, and cheap by any Western standard. The ferry from Athens to Mykonos takes 2.5 to 5 hours depending on which service you take, so combining both cities in a trip is entirely feasible and common.
See the Gay Athens Guide for sauna recommendations in the capital.
The Mykonos scene
What Mykonos does well is outdoor socializing. Super Paradise Beach during the day, the Little Venice bar strip at sunset, the club circuit after midnight. If that is what you came for, you will not feel that anything is missing. But it is a specific kind of scene, and knowing what it is in advance helps.
For the full picture, see the Gay Mykonos Guide.