This has been asked a couple of times before, but the short answer is that most sailors would have worn clothes that covered nearly all their body, even in tropical latitudes, including broad-brimmed hats. Their feet and hands would have been bare, and certainly their faces would have tanned fairly deeply, but for the most part their working clothes kept them covered and served as sunblock. For cases of sunburn, slush or grease could be made into an ointment that would provide at least a bit of relief. Here's a previous thread on the issue of sunblock:
and one with a nice answer from u/kentonj about cancer specifically:
EDIT: I'm actually going to drop in here a response to a follow up from that top thread, about what sailors wore:
Depends a bit on what your average pop culture perception is, but if you mean generally the image of pirates in billowy sleeves and striped pants, that has at least some basis in reality. /u/davidaop may be able to speak more specifically to pirate dress, but the major inaccuracies in say Pirates of the Caribbean tend to be the boots. Sailors worked barefoot, and when on land had a tendency toward cute little pumps with bows or buckles, not boots.
Check this out for more on sailor clothes and how the sailor suit developed: https://www.reddit.com/r/AskHistorians/comments/2pdr6j/how_did_the_sailor_suit_develop/
This is an image of the wounding of Horatio Nelson during Trafalgar -- it's not the most accurate depiction of the battle (the boats would be towing overboard, nasty tendency to turn into splinters if hit) but you can see a wide variety of dress among the ordinary sailors, as well as officers in blue coats and Marines in red. This is a sailor from the 1820s, so obviously later than the GAOP, but wearing shore-going rig including those cute little pumps, as is this petty officer.
