The answer is 6 years old, 6 years old, and 2 years old.

To determine the answer the first step is to produce all the combinations of three ages that can be multiplied together to get 72. Listed below are those combinations.

72 1 1

36 2 1

24 3 1

18 4 1

18 2 2

12 6 1

12 3 2

9 4 2

9 8 1

8 3 3

6 6 2

6 4 3

The second step is to find the sum of these combinations. Since the customer said knowing the sum of the ages was still not enough information then there must be more than one combination that produces the same sum. The combinations 8, 3, 3 and 6, 6, 2 both produce the sum of 14.

The last step uses the fact that the bartender mentioned his youngest child. That means the customer knows one child is younger than the other two. The customer can't figure that out with the 8, 3, 3 combination since there are two children with an age of 3. Therefore the ages of the children must be the last combination of 6, 6, and 2.