No, they do not have to match. You could do complimentary colors or carpets the same color but contrasting shades (like a dark brown with a pale beige).

Why don't you keep the basement a hardwood floor?

In my rental I used very high end carpet on the stairs with a pattern. It looks fine and it's wearing like iron! (at the top of the stairs there's porcelain tile flooring, at the bottom, wood flooring)