In my house the drywall is contact with the concrete. This is putting lot of moisture in the drywall and eventually it will start to rot. What is the best solution in this case to isolate the wall with concrete moisture. Pictures attached.
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The first issue is the moisture in the concrete - where is moisture coming from? Is the foundation sealed correctly? Even if you're dealing with drainage, you need to make sure that the basement has a proper incline (and make sure that the floor is sealed) directing the moisture towards the drain. This is a good place to start: http://www.everything-about-concrete.com/basement-floor-sealer.html
The second issue is that wood is resting on concrete. That has to change. Not just because it's kind of ugly, but also because it's going to be a repetitive source of problems associated with moisture. Exposed concrete in itself isn't terrible, but you shouldn't have any organics or drywall in direct contact with it. You could remove the wood and just have a concrete wall, or create a proper raised floor, or raise the wall with a metal separator. Painting a bare concrete wall isn't a terrible idea either - it may end up looking better than that wood, too.
I would remove the wood, seal the floor, and paint both the concrete wall and floor.