The question becomes whether you investigate as if we're looking at a criminal case, or if we investigate as if we're looking at Fortean events as a scientific hypothesis. Both would have different measures of what would be considered "proof" or "evidence". I'll start by saying I have considerably more experience of the latter!
This might sound slightly out of left-field, but for the layperson (as the majority of us are), I would recommend the recent book, Dinosaurs Rediscovered: The Scientific Revolution in Paleontology by Michael J. Benton.
It does an excellent job, not just of explaining the developments in what we know about dinosaurs and how that knowledge has grown over the past 50+ years, but crucially explains how we know.
For each advancement in knowledge, the book explains how this conclusion was reached and - more importantly - how other possibilities were ruled out.
Because that's the most important thing to remember about science and the scientific method - it is not a dogma, or a sacred text, but a constant strive not to prove, but to disprove. It's the constant revision of knowledge to encompass new theories, and the constant ruling out of possible explanations.
One of the problems in Fortean terms is that we're normally given examples that aren't falsifiable, and also tend to compound multiple assertions into one hypothesis.
For example, the suggestion that Bigfoot is a paranormal entity, belonging to another dimension or "spirit realm". In order to prove this point, you would have to not only prove that Bigfoot exists, but also prove that alternate dimensions exists, and that Bigfoot is able to access them. Where do you even begin?
You could begin by looking at physical evidence for Bigfoot - but the burden of proof is on proving that Bigfoot exists, not that he doesn't.
If science is the act of disproving null hypotheses, then we can say the null hypothesis is "Bigfoot does not exist". To disprove that, we would need proof that it does. But to approach that scientifically, you need to evaluate evidence for the existence of Bigfoot and prove that there is no alternative explanation.
The beauty of it, and the area where Forteana thrives, is that even if we say "these footprints were made by hoaxers", "this video was a man in a costume", "these eye witness accounts are from dubious sources" and "these hair samples belong to known species", the null hypothesis remains. All we have done is prove that none of the evidence presented to us is evidence of the existence of Bigfoot, we haven't ruled out that there's an unknown ape/hominid out there somewhere, waiting to be found, leaving no physical evidence behind whatsoever. We've rendered that possibility extremely unlikely, but not impossible. And isn't "unlikely but not impossible" what Forteana is all about?
I'm rambling already, but I'll add that it's not just scientific, there's another side of academia that could be well used here, and that I hope most of us are familiar with, and that's knowing well enough how to examine sources. How trustworthy they are, what the quality of research that's gone into a cited book or paper is, if there's any contrary evidence, and so on.
There's plenty of room for good old common sense, too. I'm sure we're all familiar with the concept of Occam's Razor - it's usually phrased as "the more likely explanation is most likely correct", but more accurately it's that that conclusion which requires the fewest assumptions is correct.
To go back to our Bigfoot example - if the statement given is "nobody has ever retrieved the body of a Sasquatch", and the possible explanations are "there's no such thing as a Sasquatch", "they're too well hidden", or "nobody's seen a body because they exist in another dimension"; the first requires the assumption that Sasquatches don't exist and that those who claim to have encountered them are either mistaken or dishonest, the second requires the assumption that they do exist but are capable of hiding from us, the third requires the assumption that other dimensions exists, which sasquatch are capable of travelling into, and they go there to die.
The final explanation requires far more assumptions than the previous two, which are arguably up for debate as to which is the more likely. Though I would assert that because the second explanation still relies on the assumption that sasquatch exists, that there's more burden of proof sitting with that one than with the first.
Hopefully some of that made sense! The abridged version is - both science and humanities give us a whole toolkit with which to examine and evaluate evidence and theories, and both should be applied to Forteana, along with a healthy amount of scepticism and common sense, and try not to be too blinded by that which you hope to be true.