you say it is in the basement and has no windows. So it needs effective ventilation and sounds like it does not have it. If there is an external wall that reaches above ground level, you could have a wall mounted fan. If not you could run a duct. Possibly between the joists of the floor above, otherwise it would have to go through an adjoining room, possibly concealed above a wardrobe or lichen cupboards.
I would probably go for a 2-speed ducted fan, probably running at low speed constantly until the humidity has dropped and the mould can be dealt with. With the high speed being switched on during, and preferably after, baths and showers.
If you could achieve reasonable ventilation, most rooms would be OK with a fan that comes on with the light switch and has a run-on timer, but I think a windowless basement room will need more.
Modern fans are available with very low noise, and much more powerful than the cheap fans that builders usually throw in. Ducts are usually 100mm diameter, in rigid white plastic which can be painted, but flat rectangular ones are available, and larger diameter which might be needed for greater throughput or long runs.
It is easy for a builder, electrician or plumber to bore a round hole for a duct through an ordinary brick or block wall, there is a special core drill for it. I would lean towards an electrician who can also wire it correctly.
Tell me about the room layout, and where a duct might run.