Designing Kitchens and Bathrooms - Needs exhaust Fans; Mould/Mildew

We have an older house (40+ years) with 2 bathrooms, one upstairs, one in the basement, and NEITHER have ever had a fan installed. Both bathrooms suffer from wall tile that is black from mildew and mould. The basement bath has a shower stall only with a glass door with 12" of free space above it.
Installing a fan in the upstairs won't be that difficult as the attic is just above the ceiling. However, the basement bath would require hardwood floors to be ripped-up for a ceiling fan installation. However, a fan can be installed in the wall without a lot of trouble.....but it would not be that effective because the shower stall is OPPOSITE of the wall...naturally....and so the flow of air would actually "miss" the shower stall.
Anyone have suggestions on what to do with that basement bath for ventilation ? We are ready to remediate the mould/mildew, but the effort will not be lasting unless we get some air circulation down there.

HELP ! Our bathrooms are "killing us" with mold/mildew !

Is there no way to install a fan near the shower and run the duct from the fan, out the basement wall? If you install the duct up in the joists, you should be above the sill plate and have reasonably easy access to outside. Good luck.

To do that, I've got to rip-out lapth and plaster....what a mess !
Also, I'm not sure about the floor joists above...whether or not they are running in the same direction as the duct would run. If not, then would I have enough clearance for a 4" round duct I wonder ?
I guess I could take my reciprocating saw and a thick blade and make a 6" wide cut to help make the job a bit neater/easier. With a slow speed, I've done that before with some success. It at leasts provides a notch that makes the plaster fall off in neat pieces.
Note: a local registered plumber came in and rec'd a complete re-do for the price of $15k. Now I am seeing Why !

I had a similar problem in a rental not too long ago. I placed a de-humidifier in the bathroom and ran it during showers. It looks stupid and needs to be emptied occasionally, but is better than using your walls to grow mushrooms. Also, $130 for a de-humidifier is much less time and trouble than building soffets or tearing out walls. Maybe you can build a cabinet around it (leave room for airflow) to make it look less lame.

Oh, and I found it worked better than an 80cfm fan.

what a creative approach....with incredible cost savings. No contractor I know would come-up with that one.
Thank you very much !
Now, my only problem is finding the room and the exact location to mount it.