Greg is right, start outside first. I have a new, fairly complex foundation in my home. The slope of the grade outside is correct, the exterior of the foundation is sealed, the footer has a proper French drain, and the interior of the wall is coated in spray foam, topped with a fire rated, rubberized coating. In theory it seems that it should be nearly waterproof. This past spring, after hard rains, the humidity level in the crawl space was slowly climbing, for a few days, to reach unacceptable levels. I did an inspection of everything, found that one gutter was clogged. The ground adjacent to the foundation wall was saturated and moisture was slowly migrating into to crawl space. The point being, not only do you need to control as many variables as possible on the exterior of the foundation, these potential trouble spots need to be inspected and maintained.
OTOH, it is possible to address issues from the inside, and basement water issues have spawned an entire specialty industry. They typically seal the interior surface of the wall, then install an interior drain that collects water at the base of the wall and pumps or drains it away. If you go this route, it typically involves removing the stud walls shown in your picture. If you want to try a bit of DIY remediation, first tear the wall out, then clean up the surface of the masonry. This may involve everything from wire brushing to hammer and chisel removal of loose material. At that point there are a number of products you can use to create a water barrier. I have had success with Dryloc and Thoroseal, but there are lots of options.
As for damage and structure. Unless there are very unusual circumstances present, there is no structure involved here. The pictures seem to indicate a non-structural wall, added to finish out a basement. The masonry wall supports the building, and the floor joist should be supported by the masonry, not the stud wall, which is typically installed after the home is occupied. To remove, just smack the stud off the bottom plate with a hammer, and pull it loose.