I think it depends on whether or not Alexander has a sufficient number of Companion Cavalry to attempt to pin down the Mongol horsemen. As far as I know, the Mongols relied on light cavalry and horse archers, so if Alexander could maneuver them into an unfavorable position where his Companions could be brought to bear, then the Mongols would be in trouble.
However, like comparing all great generals, there is an inherent problem in that Ghengis Khan and Alexander the Great fought with a variety of different factors; Weapons, armor, tactics, terrain. If it were an even fight, no surprises, nothing changed in respect to the two armies that they would have fielded normally, then the Mongols would win. They had superior weapons, armor, and mobility. Alexander won his battles fighting an enemy who's armies had a fairly similar composition, deployment, and tactics. The Mongols had the advantage of having fought the Chinese, the Turks, and the Europeans (for a short period) successfully. They would have had to have been extremely flexible to be so successful against so broad a spectrum of fighting styles, whereas Alexander showed his mastery against foes who fought in a similar manner and would have provided a different challenge.
Because of the huge technological differences, I would give it to Genghis. He had over a thousand years advantage in arms and armor. Alexander wouldn't have stood a chance with the tools he had. Give him the advantage of surprise, of getting his Companions into a position of attack, maybe, but it would be quite a feat.
Death is a (vastly) preferable alternative to communism.
"Idiocy knows no national or cultural borders. Stupidity can strike anyone, anywhere." -- Terikel