It is better to figure out the moisture issues ('dampness' = water) before covering up the slab. If the dampness is routinely found in one or two spots, it means there is something going on at those locations. A 'wet basement' concrete expert will offer advice.

It is cheaper to fix it properly then it is to cover it up and hope for the best. Especially if breathing or heart problems are part of the family history.

Technically you can put down carpet in a damp basement. People have done it in the past. Many continue to do it. It is not a technical problem to over come. It is a personal health concern that cause a rethink.

And the next question is: can you afford to do this twice? That is always a great question to ask yourself before trying to cover up a known concern like dampness in a basement.

Ok, thanks so much for your advice - I'll seal the slab first.

I suspect that the carpet maker would suggest some kind of underlayment as well, e.g., a moisture barrier.

I would install dri-core and then the carpet .

Short of removing the concrete floor and underlaying a new pour with thick poly or XPS, you won't remove the moisture drive (high pressure to low) that's bringing moisture into your basement. And then you would have to deal with condensation--humid warm air in your basement hitting the colder concrete.

Forget foam backed carpet tiles. Or, as noted above, create a new surface such as Dri-Core.