History of disease and medicine is a pet topic of mine.
However, I've never tried to combine it with my interest in the GAoP, so the below is informed speculation:
For the 17th and 18th century sailors, I suppose venereal disease -- syphilis in particular -- are foremost in my mind. Then there's the smallpox (measles, too) that came to the Americas on ships well before the GAoP, but was still merrily decimating native populations.
Yellow fever is another "good " one, as a mosquito-borne port disease the sailors might have encountered. Malaria, too.
Tuberculosis, typhoid and typhus, the latter two being diseases of poor hygiene...
I'm sure that the crowded conditions on a ship made the spread of lice and fleas inevitable, carrying with them typhus, etc. With so many rats on ships, rat bites and droppings in food were likely not uncommon. Personal hygiene was lacking, as was antiseptic practice in medicine, and that could not have helped matters, especially not when it came to dysentery ("flux" in contemporary documents) or typhus.
Wounds were terribly prone to infection.
Scurvy was important of course, but not contagious and therefore not that interesting to me.
Wasn't Blackbeard seeking medicine for his diseased crew? Can anyone confirm this -- or what the disease was? Aside from that story, I can't think of any pirate-specific disease-related anecdotes. There must have been syphilitic pirate captains doing crazy stuff somewhere, right? I know, not necessarily.
Mad Rose (not syphilitic)