Even though the question will never be definitively answered, readers have ample reason to believe that the ghosts truly did exist in James's 1898 Turn of the Screw. Rather than merely expounding on the evidence, I believe that an examination of motive could be revealing. The 1961 movie The Innocents, directed by Jack Clayton, portrayed the ghosts as ill at ease in their deaths. Why would the two ghosts want to terrorize Bly? What do they have to gain from it? Much, it appears.
Given the Miss Jessel (Clytie Jessop) and Peter Quint (Peter Wyngarde) died under unhappy circumstances, it follows that they would not be at peace in death either. Quint was an abusive, lecherous man who had died after hitting his head while drunk. Miss Jessel had possibly committed suicide after discovering she was pregnant. Given their behavior, it is probably safe to say that they would not be granted a place in the traditional Christian heaven. Since they were unhappy after death, they might have returned to the last place they were happy. There is an old cliché, "Misery loves company." If Miss Jessel and Peter Quint could not be happy, they would probably settle for making everyone else at Bly miserable as well.
How could the ghosts best make everyone unhappy? The obvious key was the governess (Deborah Kerr) since she was in charge of the household. If she could be made miserable, the rest of the house's residents would be as well. In doing this, the ghosts had the added bonus of driving her mad. It would appear that the two forlorn ghosts of Peter Quint and Miss Jessel succeeded superbly in their task. In the end, Miles (Martin Stephens) was dead, the governess and Flora (Pamela Franklin) were crazy, and the ghosts now had the run of the house.
In the end, the ghosts did achieve their goal. No one at Bly was at peace anymore. These wretched beings had carried out their plot to fruition. They had done so most surreptitiously as well. Driving the governess crazy proved to be the downfall of Bly.