While I totally agree with staying home if you are sick - I have to stand up for the coughers of the world. I have Reactive Airway Disease/Cough Varient Asthma. So does my Dad, so does my son, so do many other people. What this means is that anytime my airways get inflamed (could be a cold, an infection, or allergies), I cough - and I do it loudly. A cough usually lasts for weeks or months. It's gross, it's awful, but it's my life.
Cough drops do not help this kind of cough. Cough drops are for tickles. If you take a cough drop during a bronchial spasm you risk choking to death.
While obviously I do my best to keep from having this problem, and I most certainly cover my mouth, it would simply not be possible to hold a job and not go when I might cough. Mmm, maybe WELFARE or Disability would be the way to go - I'll never have to work again!
Even for regular people, coughing often lasts way beyond the initial stages of a cold. Obviously, the person in the OP probably needs to use better coughing ettiquete and know when it's bad enough to take sick leave - but a common cold isn't always a reason to skip work. Unfortunately his coworkers probably had his germs before he even became symptomatic.
I totally get that it's gross to be coughed on, but for many people coughing is just part of the package.