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.
Sounds like plaster walls and the ceiling is gypsum boards covered with just a skim coat of plaster. My house in the midwest US has a similar setup. It was during a transition period (40s-50s) from lathes to gypsum boards so the walls have lathe and then rough and finish plaster but the ceilings have gypsum boards covered with a thin finish plaster.
To hang pictures and stuff, I typically start the hole with a drill to get through the harder plaster layer and then tap the nail in the rest of the way. For heavy shelves in the closets, I've drilled holes and used those plastic inserts you tap in with a hammer and then put a screw into.
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As far as I know the more jagged and porous the material is the better. In our library, there is porous material put on walls which are faced toward the source of the sound so that it eliminates reflection. Maybe I would try eliminating those reflections between your house and the fence. But given it is in an open space I doubt it will be significant enough.
If you are very serious about it, try active noise reduction: ANC - wiki