I feel where you're coming from. Me personally if I ever introduce or refer someone I know to someone else, I will feel ultimately responsible (even if objectively speaking I had no hand in whatever transaction/discussion may occur between them).
That is called being a stand up person.
Unfortunately nowadays ppl don't have those kinds of morals.
One thing I would say is that it may be that your friend does in fact feel somewhat responsible. But sometimes friendships are like relationships where one person is more "dominant". He may not feel comfortable pushing the deadbeat friend beyond just "hey bro c'mon man just pay him." People that do this shyt once more than likely have been doing it all their lives. He has most likely did that shyt to your friend before, and might be holding off bc he assumes your friend will come to the rescue...but your friend may not be willing to do it once again.
If you actually value the friendship, maybe ask him if that's the case. 13 yrs is a long time and it may be that he's willing to give you the inside scoop.
As far as the money, obviously it depends on your financial situation, how much it is etc. If we're talking thousands then I wouldn't just let it go. If we're talking $500 then it's not worth it.