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.
They make both wall systems and flooring systems for wet basements. Sorry I am not posting links but don't feel like shopping today. A friend just put up wet basement panels (google that) and they look good and they are waterproof.
Whether this is the right solution for you depends on your house and how long you expect to be there. If your yard is graded poorly, your basement walls are in really bad shape, or there is some other intrinsic issue that will always cause moisture I would personally not do what Chris or bib suggested.
First it is a basement. How much are you willing to spend to make a place that naturally has water nice - there are "newer houses" and houses built in dry areas that don't have issues which are perfect to renovate - but that is not your house. I have seen many disappointed people spend thousands of dollars waterproofing basements to have their (newly finished) basement spring a leak the next year.
My suggestion is make the basement livable, safe, and enjoyable. Knowing that it could get flooded at some point. I would suggest putting in water-proof (rubber) planked flooring - HD sells this. Even if it is soaked it can be cleaned and reused. I would suggest some waterproof panels set about 4-6 inches from exterior walls. I would suggest no insulation other than rigid foam paneling. And if you do the foam paneling I would allow space for it to breathe. I would suggest neither drop ceiling or drywall for the ceiling - just paint it black or white (white you have to clean more).
If this extra space is really important to you - and you want it nice - I would suggest that you look at adding an addition to your house. After you do all of that waterproofing, foundation repairs, and finish your basement your $$$ is getting pretty close to adding on plus you don't have to worry about it being flooded.
Best Answer
Suggestions for investigation:
(1) Temporarily attach it to a piece of transparent tubing held vertically above the sump, to see how high the water rises. This will give you a clue to the location of the source.
(2) Force high pressure air into it and walk around listening for a hissing noise.
(3) Stop it up and see what floods or overflows. Be prepared with mops, bucket, shop-vac, etc.
(4) Inject a harmless dye or other tracer under pressure and see what turns colors. Green is best for deniability if it turns out the source is not on your property.