I've never come across any examples of pirates (and I'm going to go out on a limb and assume that you're talking about 17-18th century Anglo-American pirates here) believing in the occult, unless you count Christianity a "occult". In my researches, pirates, and other seamen of the period, don't even come across as particularly superstitious compared with later seamen. Seamen of the period sometimes held ceremonies at important points in voyages, such as crossing the Equator, entering the tropics, or passing through the Straits of Gibraltar; and buccaneer Ambrose Cowley noted that after a great storm blew up on St. Valentine's Day, it was considered unlucky to talk about women at sea (but note, this was a revelation to them, they didn't know it would be unlucky when they started talking about women). In 1719 Simon Hatley, an officer on George Shelvocke's privateering voyage of circumnavigation, shot an albatross and the event was used by Coleridge in The Rime of the Ancient Mariner, from which we get our modern superstition that killing an albatross is bad luck. In the original, though, the albatross itself was believed to be the cause of the bad luck, and Hatley shot it to bring their bad luck to an end - actually, that's probably one of the best examples of literature obscuring the historical truth, in this case completely reversing it. As for pirates, I've never seen any reference to curses like or unlike the black spot, or indeed any other occult-type practices. In Thomas Anstis' company, unwilling new recruits were made to sign the ship's articles in front of a mirror, which the late great Ken Kinkor believed had occult significance, but it was just as likely, perhaps more so, that the mirror wa there so the signatory could see the two men with loaded pistols standing behind him, and Kinkor, for all his greatness, believed a lot of thing about pirates on the slenderest evidence. (The account of the practice, by the way, can be found in the 1724 trial of Bridstock Weaver and William Ingram). Other than that, any mention in contemporary source of the pirate beliefs suggests overwhelmingly that they were essentially conservative Christians. Some members of Sam Bellamy's crew hated Presbyterians. Given that pirates were, on the whole, just people, who came from and maintained links with regular society, it would be surprising if their beliefs differed markedly from anyone else's.

Since the poster of the question clarified what they meant by occult, and /u/ETFox pretty much said what I would say on the sailor/pirate would or would not believe, I'll address a further aspect that kind of came up - African beliefs that might be interpreted today as Voodoo. What we call "voodoo" today in popular culture wasn't the thing we are familiar with today until the 19th century. The word voodoo didn't even enter the language until the 19th century. Honestly, much of what we think of as "ye oldie myth and folklore" really comes from the 19th century when people really buckled down and began writing down local myths, folklore, songs, and legends (part of it might have been the fear that if they didn't write it down, with industrialization, the old stories might be forgotten).

So, some might be wondering, why pirates have this current association with voodoo. Why does Jack Sparrow and the gang encounter it so much? In the 1960s, Disney made the Pirates of the Caribbean Ride in Disney Land. Since they wanted that American feeling to things, they included that Louisiana bayou twist to it in the beginning. After all, a 19th century pirate, Jean Lafitte had operated from there (though people don't realize there's a difference between the "Golden Age of Piracy" pirates of the 1690s-1720s and those of the early 19th century). Tim Powers, one of a group of writers that would really help fuel the modern steampunk movement, wrote a book on pirates that mixed in voodoo in 1987. Powers had been inspired by the Disney ride while writing his pirate book, On Stranger Tides. This book helped kick off a rise in pirates being mixed in fantasy. On Stranger Tides inspired the Monkey Island game franchise in the 1990s. This franchise in turn had heavy influences on inspiring the plot lines of Disney's Pirates of the Caribbean franchise (which is also based on the original Disney ride). Disney's franchise also picked up the book On Strange Tides to inspire their 4th movie - and mind you the book inspired the game that inspired the films, and yet the ride that inspired the movies also inspired the book On Stranger Tides that Disney was using to inspire their 4th movie....Yea, it's confusing that all these inspirations are getting crossed and practically creating plot inbreeding. But, as a result, pirates are now well tied with voodoo in our current pirate popular culture.

But let's address the African belief themselves. If you really want to get to know what Africans who came over here as slaves believed religiously in the early eighteenth century, I recommend two books:
Sweet, James H. Domingos Álvares, African Healing, and the Intellectual History of the Atlantic World. Chapel Hill: University of North Carolina Press, 2011.
Sensbach, Jon F. Rebecca's Revival Creating Black Christianity in the Atlantic World. Cambridge, Mass: Harvard University Press, 2006.

I've read both those books, and let's just say it's not simple and it's much more than I ever expected. The first book is about early 18th-century Brazil and a African medicine man there. The other is set in the Danish islands in Caribbean during the early 18th century and involves a mulatto woman who was a former slave. Tons of topics I had known nothing of or little of before. Both are good for broadening your horizons.

Would you mind explaining what you mean by the "occult"?