Just like Polly stated, it sounds like you have two problems to fix here: The salting of the concrete (the cement breaking down) and the repairing (replacing) the dry wall (gypsum board).
You likely have a moisture problem here as the two walls create a place for the moisture to collect.
If this wall is below ground then you could be facing water penetration through the basement walls. Make sure your rain gutters are aimed away and have at least 12ft of distance from the house so water does not collect near the foundation. You may need to seal the foundation by digging around the outside of the foundation wall to seal it with tar (water emulsified not solvent) and if you can put Tyvek (polyester) fabric (in a warm climate) or Styrofoam foam board (in the cold climate) along the outside to stop the water penetration. If you have any cracks you will need to scrape out any loose mortar & re-mortar if it's large, or use foundation chalking making sure there is no loose dust in the crack.
As for the inside concrete wall you'll need to get some muriatic acid to clean the wall after you have used a wire brush to remove any loose material. Then wash down the wall with water and the acid and water again.
Make sure you wear proper protection here breathing the dust is not healthy and protect your eyes as well. With the acid you need proper gloves and body covering and need good ventilation (open doors, windows & fans).
Once the wall is cleaned you will need to patch any cracks like you did on the outside. Then you will need to get basement & masonry waterproofer paint to seal the wall If the floor is damp you will likely need to treat it as well.
Once the foundation or external walls are treated you will need to give them some time to dry. You should notice less of a musty smell and less humidity. If you can wait until the late spring in sealing the walls and replacing the dry wall (to make sure you don't have any leaks).
Given what you stated here it's likely you have mildew and/or mold (black or orange spots on the back side of the drywall and/or wood studs). If you see it or smell it you will need to remove all of the dry wall and sanitize the area fully (of course you will need to do this first before doing the external wall treatment).
You will need to use proper gear here as well as you don't want to get sick from the mildew or mold (Tyvek suit, full face goggles & the proper respirator for mildew & mold).
Once you remove all of the dry wall in sealed plastic bags fully vacuum up all of the loose dust. Then wash down with diluted bleach the wood studs and the floor. You may need to hit the ceiling as well, to kill any remaining spores.
If this is below ground I would replace the dry wall with a PVC/fiberglass wall system for basements. I would also insulate the wall space between the walls with open faced fiberglass (no vapor covering) if you live in a cold climate.
If this is an above ground wall you can use a closed face fiberglass or spray on insulation with a full vapor barrier on the inside wall surface you will be putting the drywall on top of (no gaps) as you don't want the inside moisture to condense in the space between again (warm or cold climate). Then apply your drywall (gypsum board) with fiberglass tape and dry wall mud to seal the gaps between the sheets and two or three coats of paint.
Don't skimp here as you don't want to do this again and you don't want to have an un-healthy house either (mildew/mold).