I saw your post but initially assumed that you were talking about diseases in fish that had been handled. I had never heard the term, fish-handler's disease before, but there are definitely bacterial infections which can be transmitted to humans. I have not seen the article you describe. Most of these infections are caused by different species of vibrio bacteria (esp. v. vulnificus and v. parahaemolyticus). I have no idea how frequent these are. I have never seen a case before (I am far from saltwater in WI and there are only 2-3 tanks in the county). Some of these can be severe but I believe most are mild.
Based on the lack of threads on this subject, I don't think it is very common. I put my hands in the water (as long as they are clean, I am usually more worried about me hurting the fish (soap residues) than I am about the fish hurting me). Usually saltwater in an open cut burns enough that you stop further exposure. However, I would make sure that your physician knows about any nonhealing wounds on the hands and knows that you have a saltwater tank. Tetracyclines seem to be the best treatment which is different from many other typical skin infections.