Why doesn't Bailey's go bad? I realize it can go bad after a long time, but why does it take so much longer than other dairy products? Does it have to do with the alcohol content? If so, what is the process that is happening?
Here is a quote from the company that makes the registered trademark Baileys Irish Cream:

Baileys® is the only cream liqueur that guarantees its taste for 2 years from the day it was made, opened or unopened, stored in the in the fridge or not when stored away from direct sunlight at a temperature range of 0-25 degrees centigrade.
One of the keys to achieving this 2 year shelf-life is in our patented process of blending of fresh Irish cream with the spirits and the whiskey without the use of preservatives. The alcohol acts as a natural preservative for the product.
Under normal conditions of storage Baileys has a shelf-life of 30 months.
If you are concerned about a bottle of Baileys® please check the best consumed before date on the bottle - all bottles now carry a best before date. This number is located on the bottom left hand side of the back label. Example : Code 11 20XY would mean that we guarantee the product would taste perfect until that date (XY is the year 2 years from the date of manufacture)

(source).
The alcohol content is the sole preservative listed.