Rubber flooring like in the free weight areas of fitness centers

That's an awesome idea! Could I put that right over the basement floor or would I have to put something under it?

rubber is a good idea, and yes it can go right over concrete. My suggestion is look into a product commonly used in gyms, and indoor sports centers called MONDO.

Here is a link that might be useful: Mondo flooring

Professional gym quality and performance for the home. We are the direct manufacturer. Advanced Home Gyms for total body training at affordable prices.

Here is a link that might be useful: Home Gym