I mean they did bring an end to the Islamic civilizations in the area, which very much influenced how Islamic and Christian relations would play out for the next couple hundred years.
Also slaughtering millions of people at a time when the total population of earth was still far far below a billion tends to leave ripples.
They didn't exist for more than a couple generations, and really only with Gengis as a global power, but they did have a massive influence if purely because several powerful places on the map simply ceased to exist after they meet the mongols