I realize that the burden of proof falls squarely on the shoulders of the person making the claim, but they often seem unwilling to do the footwork.

It is difficult to prove a negative case except by pure logic, so once that avenue is exhausted would it be worthwhile to attempt to make the best positive case to show that it still fails to pass muster? This seems to be the way Mythbusters tackles many difficult claims, however 'true-believers' still seem unlikely to respect their methods.

Example (for the sake of argument): Bigfoot is an extant creature living in North America.

Are there any other alternative methods to prove a negative claim?