I was under the impression that Genghis Khan was an invader while Kublai was more of a patron of fine arts; but reading Wikipedia it mainly mentions his campaigns for the Mongol throne and the unification of China.
Did he actually contribute significantly to world culture, and if so in what ways?
The most important thing Kublai Khan did for culture was to found the Yuan Dynasty, which sought to rule what Genghis had conquered. It initiated trade between east and west, cross-pollinating ideas and culture. Specifically in China, this resulted in advancements in the arts - painting, calligraphy and poetry combined into a new discipline similar to Persian art, and poetry likewise was introduced to theater, along with western instruments, in zaju and sanqu.
The cultural impact of Marco Polo is strongly linked with Kublai Khan as this was who Marco Polo was said to have visited as a high point of his Asian journey.
What the Mongols may have taken in lives and bubonic plague is perhaps offset by a long step forward in international trade and globalisation; or at least accelerated thereby.