Why doesn't Bailey's go bad?
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.