This is something that interests me, as a matter of fact; one of my other, relatively harmless hobbies is music.
Sea shanties are an interesting historical topic; they reveal a lot about the culture and behavior of seafaring people over the years. In the past few decades scholars have studied them pretty thoroughly. The problem, of course, is that the real things were seldom written down or published until recently; they were passed around from sailor to sailor and from generation to generation. (I have no idea how many different verses of "What do we do with a drunken sailor" have been written down over the centuries, but I imagine they only represent a fraction of the total.)
There are quite a few recordings of sea shanties on the market, ranging widely in quality and style. Musicians wanting to record them have several options. An "authentic" recording of a sea shanty isn't necessarily the most pleasant thing to listen to; the original "musicians," after all, were amateurs in every sense, and usually had no interest in what anybody other than themselves thought of their "performances." (And the lyrics ranged from the silly and naive to the obscene.) Several modern groups specialize in performing, and recording, shanties as they "actually sounded." I'm glad those groups are out there, I enthusiastically support what they're trying to do, and I always take the opportunity to listen to their performances. But I have to confess that I don't take much pleasure in listening to their recordings repeatedly.
On the other hand, one of my all-time favorite recordings is the famous one by "The Men of the Robert Shaw Chorale," from 1960: http://music.barnesandnoble.com/search/product.asp?WRK=3103240 . I bought it on LP about thirty years ago and wore out the vinyl; later I got the cassette tape version and kept it in the car. The CD version now lives in my workshop. These performances have nothing to do with what sea shanties sounded like when sung by sailors on board a ship. The songs have been gussied up with modern, four-part arrangements, and the Robert Shaw Chorale was one of the finest choral groups ever; there's just no way any ship's forecastle ever produced a sound like Shaw's tenor section. By the standards of authentic performance practice, this recording is a joke. It's also one of my favorites - and apparently lots of other people like it too. That link contains some sound bites; take a listen and I think you'll see why the recording is so popular.
For anybody wanting to take the plunge and try some "classical" music that has to do with the sea, I can suggest the following as good starters:
Claude Debussy, "La Mer"
Richard Wagner, Overture to "The Flying Dutchman"
Arnold Bax, "Tintagel"
Ralph Vaughn Williams, "A Sea Symphony" (No. 1 in C major) and "Fantasy on British Sea Songs"
Benjamin Britten, "Four Sea Interludes from Peter Grimes"
Maurice Ravel, "Un Barque sur L'Ocean"
And, of course, any discussion of this subject has to include the score by Richard Rodgers and Robert Russell Bennett for the TV series "Victory at Sea." The stereo studio recordings of some of the music are available on CD: http://music.barnesandnoble.com/search/product.asp?WRK=152216 and http://music.barnesandnoble.com/search/product.asp?WRK=149683 , and of course the whole series has been released a number of times on DVD. The DVD version I have is a perfectly satisfactory one that I picked up at Wal-Mart. It's packaged in four "volumes" of one DVD each, which Wal-Mart was selling at the time for (drum roll, please) a dollar apiece. Again, those Barnes and Noble links include sound bites. They also include a couple of snotty reviews by a critic who ought to be strung up by his - well, never mind. Whatever else may be said about it, it's great music to build ship models by.