Behind every beautiful thing there's been some kind of pain"
In this secular3 age where questioning, and doubting are the name of the game - it seems one of the most prominent results is that some kind of dirt (the 'pain' of which Dylan could be speaking) has been dug up or made up about literally everything you can believe, and it seems that all the Abrahamic faiths (Christianity, Islam, Judaism) have got it worst, but things like Scientology and even Atheism have certainly got their 'skeletons in the closet.'
Each faith sets forth what is intended to be a beautiful image of existence, present or future - and a beautiful set of principles by which to achieve the best possible existence. But there is always some dirt to smear in all of it... so I suppose what we're all looking for is the faith/nonfaith that doesn't stain. The immanent focus of this age is prone to staining, after all, nothing in the material world is perfect, no man or woman, no society, no creature, even nature itself can seem a little mucked up sometimes... but if we can use our imagination - our spirit's imagination, we believe there must be something higher (not necessarily transcendent), something which does not stain at all, but it is so hard to find because you gotta clean the dirt off anything before you can tell if it'll stain... so we feel the Malaise.