How to respond to an upset student?
I would not intervene unless I felt the student was at risk of harming themselves or others.

As you say, college is about more than grades, learning how to handle failure is part of life. 

Moreover it’s not unlikely that their performance on the final was the icing on the cake: they had a bad day/week and were upset about all of it plus the final.

Unless you’re willing to allow them to retake the exam (which is probably a bad idea unless they offer a good reason), reaching out to them and telling them they should deal better with exams will likely annoy them and demotivate them further.