It all depends I think. We all know the type that is needlessly offended, quick to lose their cool, generally difficult to deal with. Apologies may not be in order, they are going to have to learn to grow up.
With that said, I still swear by how to win friends and influence people, especially in the work environment. One of the keys in that book was when you make a mistake, own it, and apologize. I think that's true at work, people will be more apt to work with you when they realize you are honest with yourself and not quick to shift the blame.
Just my 2 cents.