In the book, The Lord of the Rings, Tolkien often talks about the central concept of the Rings of Power, namely that the Great Rings preserve life.
Gollum, for one, is incredibly old, having possessed the Ruling Ring for most of the Third Age. This makes him anything up to about three thousand years old, although we do not of course know a precise date for when he acquired the Ring, only that it was found by him a long time after Isildur was slain and the Ring had been lost in the Great River.
Gollum has not faded, despite possessing it for so long. Bilbo - who also possessed it - talked about feeling thin and stretched, a sign that the Ring was beginning to take control of him; yet he, like Gollum, showed no actual physical signs of the fading which was one effect of repeated use of the Ring's power to render mortals invisible.
The Ringwraiths, however, do show that fading: they cannot be seen by mortals now. When Frodo encounters them at Weathertop, in 'The Fellowship of the Ring', he can only see them when he puts on the Ring. He then stands at the threshold of their world, and the wraiths who inhabit that world are revealed to him. Ordinarily, in our world, the wraiths have to wear cloaks, boots and other clothing in order to give form to their nothingness.
But everything that Gandalf reveals to Frodo - and to the reader - in the chapter 'The Shadow of the Past' - implies that a mortal who possesses one of the Great Rings does not die, although neither does he obtain more life -- he simply continues, until every moment is a weariness. And he may fade, if he repeatedly uses it to become invisible.
But the Ringwraiths seem to be in a peculiarly uncertain state: the very name, wraith, implies that they are dead, in other words that they are spirits of the dead, and thus that the Nine are the spirits of those mortal men to whom Sauron gave the Nine Rings (Nine for mortal men, doomed to die it says in the rhyme), and so he enslaved them.
Yet the idea that they are mere spirits of the dead seems to be in conflict with the concept of the Great Rings conferring long life, and especially with Gandalf's statement that a mortal who possesses one does not die.
If we look further back, to the pages of The Hobbit, we find the earliest conception of Sauron, whose true identity is then still concealed from Gandalf and from the reader: in the earlier book, he is exclusively referred to as the Necromancer.
Although Tolkien nowhere in that tale expands on his use of this term, any good dictionary will tell us that the term necromancy means the use of magic to communicate with the dead. Thus in the earlier tale, Tolkien is implying that Sauron is a magician whose principal business is to commune with the spirits of the dead. This ties in with the later use of the term ringwraith, linking Sauron with those of his servants who are wraiths - and, as defined by the dictionary, this term, too, refers to spirits of the dead, and is entirely consistent with the term necromancer.
Although it is somewhat negative logic, one has to ask: why is Sauron called the Necromancer? He must be in communication with the dead, which is what the term implies; but who, save for the Nine, is he in touch with who are dead? If not them, then who? I cannot recall any other characters with whom he has dealings who might be dead, yet he must - logically - have dealings with some who are, else why does everyone call him by that name?
This thus brings us to the central dilemma: are the Ringwraiths really alive?
For, on the one hand, as mortals who possess one of the Rings of Power, they are said to be alive, just as Gollum is alive, who also long possessed one; but on the other hand, the use of the term wraith implies that they are merely the spirits of the dead, akin to the Men of Erech, something which is reinforced by the use of the term necromancy in relation to Sauron, whose servants they are, and who has used the Nine Rings to enslave them.
Tolkien seems to make no clear distinction between the living and the dead, given that the Nine appear to be treated in some respects as though they are alive, yet in other respects appear to be treated as though they are dead.