When I was looking for the definition for "airag" the other day, I found this website: "All Mongolian Recipes: The Food of the Nomads."
As one might suppose, this website describes what Mongolians eat and drink today. For example, today's Mongolians enjoy vodka, which they obtain by trade with the Russians; however, vodka was not available to them in Genghis Khan's time. However, a lot of dishes still eaten are "traditional", and by the description of how they were made, likely were eaten in Genghis Khan's time.
Airag, by the way, is a mildly alcoholic brew made from mare's milk. English speakers may be familiar with this item under its Russian name, kumiss. The web site describes its manufacture in fascinating detail here; I commend this page to readers interested in kumiss, or in unusual alcoholic beverages in general.