We had a similar problem with our basement walls.
Moisture easily travels through concrete, so Step #1 is to look for and divert sources of water on the exterior
- e.g. gutter downspouts - into drain pipes or dry wells. As others noted, the junction between the wall and sill plate can also be vulnerable, particularly if you live in an area with snow, or if the landscaping / soil level outside slopes toward the top of the wall. If landscaping / soil slopes toward the wall, you should also install dig a small trench along the wall that is ~1 feet deep and 1' wide,, put a couple of inches of ~3/4" rock in the bottom, then install a perforated drainage pipe at the bottom, wrapped with landscaping fabric (to keep dirt out), then fill the trench to the top with 3/4" rock. Home Depot sells premade black plastic drain pipes like these, wrapped with fabric around a layer of styrofoam beads. Run the drain pipe to a lower slope or a 'drywell'. Home Depot sells dry wells too. If the soil in your area is clay-like and less permeable, you may need to dig a bigger hole for the plastic dry well, and surround it with crushed rock to increase the surface area.
Step #2 is to block moisture from coming through the wall.
If it is poured concrete vs concrete block, this will typically be much easier. First chisel off as much of the existing material as possible -- particularly around the loose area. Look for cracks in the wall. If you find one, fill it with Sika Concrete Crack Fix. See: Sika® Concrete Crack Fix
Then make sure the surface is fee of all loose material and paint it with Drylok waterproofing paint. See: DRYLOK® Masonry Products: Paint, Sealer, Concrete Waterproofing
If cracks are large or the wall is concrete block, sealing becomes more difficult. Our basement walls were concrete block with leaks in the corners and the junction with the floor slab. I bought a kit from Foundation Armor, which worked like a charm. See: Concrete Repair-Wet Crack Repair
You will need a hammer drill (not battery powered) to drill a series of holes along the crack, to pump single-component polyurethane foam into the holes. Like Great Stuff canned foam (also single component) the Foundation Armor foam cures on contact with moisture / water -- so when it encounters water it rapidly expands to fill and seal the crack. It can even be used on wet, actively seeping cracks. This product took some work but solved the problem. Then I power wire-brushed the walls and painted them with two coats of Drylok. Problem solved.
After you paint the concrete wall with Drylok, you can apply standard stucco or cement / mortar, which would be more resilient than plaster. We have a home in France with thick stone walls, and the surface is smooth mortar. A skim coat of plaster is applied last, then paint. If you decide to use plasterboard / drywall, make sure you buy "greenboard" with is used in bathrooms etc and is more moisture resistant. A better choice would be 'backer board' for tile which is waterproof and used in baths etc. This can also be coated with a skim coat of plaster for a finished and smooth wall surface.
I hope all of this info is helpful.