Wall to wall carpet may be the worst choice for a basement but people do it because it makes the space feel cozier, it provides a soft landing for playroom falls and it's affordable. Having experienced floods in two carpeted basements (an urban duplex-down and a suburban single family home), I can safely say that I will never install conventional carpeting below grade.
If you love the look and feel, go for FLOR or get some huge area rugs. It's easy to create the illusion of wall to wall carpet with custom-bound rugs made to fit your space. If cushioning for kids is your biggest concern and you're considering interlocking foam mats, do your research to be sure they don't contain harmful chemicals as concerns have been raised regarding their toxicity in recent years.
Wood, cork and bamboo are not good choices for basements because moisture will destroy those materials. Ceramic and porcelain are easy to clean and maintain, and can be made to resemble other materials including wood and slate. Marmoleum is one green option that is durable and versatile. It can be put down in tiles or sheets.
So, what's in your basement?
(Image: Marmoleum/Forbo Flooring Systems)