Basement – Should I frame out the foundation walls before or after I install the subfloor

Water IS getting to it. There IS water there. You cannot fix it by covering the wet. You must fix it by stopping the water from getting to that point.

Check your gutters. Do they tend to get clogged? Are the downspouts set so the water is directed away from that area?

Check the grading in that area.

If all of this checks out, then the water is getting through the wall. There may be a crack. You may need to dig down and apply water proofing in that area, or repair the crack.

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.