Great question.  I learned about this when I was a kid and tried it myself.  It gave me some relief, but not enough to matter. 

The truth is, coffee is a mild bronchodilator.  It was actually used by asthmatics back in the 1800s and earlier to get some asthma relief.  It was one of the options Teddy Roosevelt used when he was a kid back in the 1870s.  It wasn't a great bronchodilator, but it did give some relief. 

Coffee is a member of the Xanthine family the same as Theophylline is.  Theophylline went on to become a front line therapy for treating asthma from around the 1950s until the late 1990s.  It was actually used more as a preventative medicine the way Advair is used today.

I would not recommend using coffee as an asthma medicine.  You are better off keeping a Ventolin inhaler on hand to treat acute asthma episodes, and to prevent asthma with controller medicines your doctor will prescribe.