How much does a basement cost?

It cost around 10-25k, depending on size and current condition of a basement. 10k would be on the lower side, and in most cases it would cost more. Although, this would include just the finish basement and basic electric stuff (outlets, lights etc) if you want Plumbing work (like a bathroom) or ac/heating it would cost more.

And if you buy and then do a finish basement, you have to pay for it, if you buy a house with finished basement then your downpayment will only increase slightly. Still, you should buy a house you like. Finish basement isn't hard to do, as long as you can pay to get it done. BUt the main thing is that to find a good quality experience basement renovation or construction contractor. That will help you much.