Far too often

…and by the Name ABADDON which Moses invoked and sprinkled the dust towards heaven, and immediately there fell so great rain upon the men, cattle, and flocks, that they all died. (cf. DeliriumsRealm.com)

What do we mean when we say a person is possessed of demons?
Once upon a time, it was meant literally (cf. Mk. 5:1-20).
Today we talk of a person tackling their demons in a metaphorical sense. GoodTherapy.org has a short list of demons with which we struggle today. My own personal demon, in this sense, is anger. Does it help to be able to say I have anger issues? Not in the slightest little bit. The words just don't fit the crime.
On all my school reports, from 1973 through to 1978, my teachers would say I had a bad temper. These days, of course, we can't say a student is in any way 'bad'; we must concoct word puzzles to express the obvious.
I still have a bad temper.
I find it unfortunate that, when I was younger, we had the words but not the method. Today, they have the method but not the words.
I suppose now we would say that both Mulder and Scully have 'issues'. In large measure, the X Files is about these two individuals working out these issues. Neither one of them do, though. Personally, I think Scully does better than Mulder.
The difference, I think, is that Mulder died and was reborn (cf. 'The Blessing Way'). Scully, on the other hand, had to deal with death on a very human level. From 'Leonard Betts', in Season 4, through to 'Redux II’, in Season 5, Scully was dying of cancer. Of course, she was never going to actually die, for there would be no X Files without her. Despite it all being about Mulder in the beginning, I still believe that his disappearance was never crippling. Without Scully, there would be no X Files.
Maybe it comes down to something as simple as Scully being female. There is little doubt that she had a feeling of obligation to Mulder, in the same way she had to her father. She admitted as much in 'Never Again':

It usually starts when an authoritative or controlling figure comes into my life. And part of me likes it, needs it, wants the approval. But then at a certain point, along the way, I just, you know…

Along the way, and certainly facing cancer must play a part, she learned somehow to deal with her feelings. Not in any concrete sense. The beauty of these characters, the reason we keep watching, is because they remain human. Scully would continue her fight against her demons, a fight we could see her winning, or making progress at least.
Mulder was always lost. Nothing was ever going to save Mulder. Except Scully.
At the end of 'Demons', Scully, even though he had a gun pointed at her, never gave up on Mulder. She had more faith in him than he ever had in himself. Because she understood that what he had done to himself was out of guilt. Why did Mulder choose this moment to subjugate himself to such extreme therapy, to put his life in danger at the very moment when Scully needed him to be strong? Because it was his fault she was suffering and there was no way for him to deal with this reality except to put himself in the same position. It was his way of acknowledging that she suffered because of him and all he could do was to suffer as well.
The strength of the woman is that she understands the weaknesses of the man. Scully can be strong enough for both of them.
Mulder's demons stem from the fact that he isn't a traditional alpha male. He empathizes with victims, understands what they experience, dedicates himself to finding solutions to their problems. Mulder is selfishly selfless.
We love him for it. As much as he can be obnoxious, we understand that he is because of his ability to care so deeply.
Scully cares just as much without losing her place in reality. Scully, as will be shown and as I've mentioned previously, is maternal. A mother must care in a way that will ensure the survival of her child. There is a pragmatism to a woman's caring that men will never understand.
Mulder has to put himself in harm's way to understand that his partner, because she is female, lives in harm's way.
If nothing else, the X Files certainly makes us aware just how dangerous it is to be female in the modern world. And their demons are, far too often, the men in their lives.