Assess the damage:if moderate, try removing the mold yourself. Use fans to move the air and flood lights to see and plastic covers to collect the mold when scrapped clean. Use liquid spray that you can buy at the hardware store.Some suggest surface blasters. After the mold area is clean and dry put in a dehumidifier and check for ventilation as preventative measures.
If it's ground water, the issue isn't water vapor, but rather just plain water. Is the water table close or higher than your basement slab? If so, not a whole lot you can do to completely stop moisture issues, as that basement was just built in a bad spot.
That said, the main solution would be a sump pump and drainage tile system. Any water coming up through the earth dumps into the drainage tile, into the pump well, and gets pumped out. Again, though, if the water table is already higher than the basement slab, that'll be a never ending battle as well.
That said, perhaps the issue isn't as much about ground water as you think. Does it get humid in your region? If so, then 'damp' basements are par for the course. Basement walls are usually always going to be cooler than the air, and, a such, will be were condensation forms in a humid environment.
To prevent that, you need to a) dehumidify and/or b) insulate the walls.
A dehumidifier constantly running tends to be a normal part of any basement in a humid zone, so that's a good start. Insulating with walls with XPS or EPS foam will help too, it'll keep the moist air further away from the cold wall.
I can't say if an air exchanger would help or not. If it's including an A/C system, it could help (as the A/C is a dehumidifier) but otherwise I have a hunch you'd just be pumping more humid air into the space.
Products like drylock don't do a whole lot. They're not strong enough to prevent hydrostatic water pressure (high water table) and do nothing to prevent condensation. They're a bit of a gimmick.
Best Answer
If you're building a proper sauna (Finnish sauna) and using it correctly, there should be no moisture problem in it. The air you feed it initially has the same humidity as the surrounding air, then the stove raises the air temperature leading to a decrease in relative humidity (same water content by weight + higher temperature = lower relative humidity, RH%).
The only two things that can add water vapor to the air in your sauna are 1. multiple people sweating in there for a while and 2. somebody pouring unnecessary amounts of water over the heated rocks, thinking a sauna should look like the steam baths (Turkish baths) they've seen in the movies, where people can barely see eachother through the thick clouds of vapor.
People in a sauna still need oxygen replenishment, so you should build it with a fresh air intake under the stove or somewhere close and low, and an air exhaust on the opposite wall and at half height or higher (sometimes it's a small sliding wooden window on the ceiling). This should deal with the humidity from sweat just fine.
Stop it. A sauna is not a steam bath. The air in there should be (very) dry. The only time you pour 1-2 ladles of water over the rocks is maybe if you want to "kickstart" your sweat response by generating a sudden wave of extra heat, or if you're looking for the supposed antioxidant or other benefits of löyly. Other than that, adding too much water just makes the high heat unbearable, shortens your sauna session to way under the normal 15 minutes, plus it introduces unecessary risks of germ growth, which is normally inhibited in a sauna by both the very high temperature and the dry air.