First, I would say they do exist just because of the detail the governess went into when explaining herself to Mrs. Grose. The governess knew certain things that would be impossible to know unless she actually saw the ghosts. Also, the governess seemed like such a credible character in the beginning that I found it hard not to trust her. Mrs. Grose seemed to trust her to the end, even though she still could not see them for herself. This is evident when Mrs. Grose agreed to take Flora away from the house. Another thing that convinced me that the ghosts did in fact exist was the strange behavior of the children. I would think it impossible for two young children to behave as well as they did. The last thing that convinced me that the ghosts did in fact exist was the 1950 play The Innocents, by William Archibald. In this particular adaptation of The Turn of the Screw, Archibald makes it very clear that the ghosts are real. However, in the 1961 film, The Innocents, directed by Jack Clayton, the existence of the ghosts becomes somewhat problematical again.
Then, after reading the book again much more carefully, I gathered my senses and came up with a rational explanation. I remembered the short story "The Yellow Wallpaper," by Charlotte Perkins Gilman, which was a story about a young lady subjected, by doctor's orders, to not under go any activity whether it be physical or mental. The doctors called it the "rest cure." At one time it was a very popular diagnosis; but it led many women, like the author and the narrator, to drive themselves insane. The narrator eventually drove herself mad from not doing anything except imagining living beings lurking in and behind the wallpaper. Likewise, I think, since the children in The Turn of the Screw and The Innocents were so over-well behaved, that the governess, named Miss Giddens in the play and movie, as played on screen by Deborah Kerr, did not have much to occupy her mind with. Thus, after a while, when the day's consistent routine became too much to bear, she started having illusions that manifested themselves into ghosts she claimed to see. Therefore, she basically went insane by her own mental inactivity, which led to the destruction of her rationality.
Although it can be argued for a lifetime whether the ghosts existed or not. It is up to the reader to decide. I feel Henry James's book The Turn of the Screw is a masterpiece for that reason alone. The question of the supernatural is one that will continue forever. Many adaptations have followed The Turn of the Screw; the most popular is the 1950 play The Innocents, by William Archibald, and the 1961 film The Innocents, by Jack Clayton.