I see that a lot of people have recommended linoleum. I agree that would be a very good choice--although it will probably be cheaper and easier to find vinyl flooring. Linoleum has certain properties that purport to make it non-allergenic. If I remember correctly, it also has anti-static properties. You can create unique designs if you have a skilled installer to make "inlaid" patterns. Of course, vinyl now comes in all kinds of attractive patterns as well.
Cork is a great option, but it can get expensive. The benefits being that it is a little warmer and forgiving if you drop something. It can dent, although some denting will rebound over time. It is also renewable and eco-friendly. And may be hypo-allergenic as well.
Carpet is the warmest option--sometimes important in a cool basement. But if you plan to let your kids do craft projects on it, it will obviously be more difficult to keep clean.
Have you looked at the rubberized tiles?
A last option is to stain and seal the concrete floors and use throw rugs where needed. I have nearly 5000 sq ft of colored and polished concrete in my home. We love it--it was stained the color of rootbeer and looks a little like leather tiles. (Leather, another option, but not very practical for a child's play area). I wax it once every two weeks or so.
Good luck!
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