If no, how do we check if a journal is predatory?
I don't think we need any additional policies about referring to or questioning about papers published through predatory journals.
Assuming the paper is mainstream...
Well, we don't really need to do anything. Maybe, in the case that the post is written by the author of the paper, we could provide a link indicating that journal is predatory, but nothing really needs to be done here, treat the post as any other.
Assuming the paper is non-mainstream...
We already have functioning policies on questions about non-mainstream physics, so I think this would be handled already. We also handle answers based on non-mainstream physics with downvotes (and in some cases deletion).
We also close questions based on peer review, regardless of the mainstream/non-mainstream aspect, so this shouldn't be a concern as well. Hence, I don't think an additional policy is needed.
This Stack Exchange is about "mainstream physics". It is about mainstream physics whether it is published in leading journals or trailing journals. It is not about non mainstream physics whether the non mainstream physics is published in leading journals or trailing journals. This Stack Exchange is about mainstream physics, not journals.
Anyone can easily find non mainstream physics articles published in leading journals. And anyone can easily find mainstream physics articles published in trailing journals.
Why would anyone publish in a trailing journal? I will tell you a story.
Some years ago, I discovered a new method of writing unitary matrices. I messed around with publishing it at Journal of Math Physics but they pissed me off. I'm an amateur. I do not need to "publish or perish". I do not need to have other physicists read or care about my work. So I put it on to Vixra here where it has presently been downloaded 1082 times:
Unitary Mixing Matrices and Their Parameterizations (2015) We present a new decomposition of unitary matrices particularly useful for mixing matrices. The decomposition separates the complex phase information from the mixing angle information of the matrices and leads to a new type of parameterization. ... http://vixra.org/abs/1511.0083
A new decomposition of unitary matrices is an important and unexpected addition to quantum mechanics. So it has been picking up citations:
Parametrization of 3 × 3 unitary matrices based on polarization algebra Gil, J.J. Eur. Phys. J. Plus (2018) 133: 206. https://doi.org/10.1140/epjp/i2018-12032-0
My paper also gives a geometric calculation for CP violation and is cited here but they replaced "vixra" with "arxiv" in the text:
The Geometric Origin of the CP Phase H. Fanchiotti, C.A. García Canal, V. Vento The complex phase present in CP-violating systems such as neutral kaons is shown to be of geometrical origin. It is also concluded that the complex phase of the Cabibbo--Kobayashi--Maskawa (CKM) matrix is a Berry-like phase. https://arxiv.org/abs/1705.08127
The above paper isn't published (yet), but it is good enough that it is cited in a peer reviewed paper here (as ):
Testing CPT violation with correlated neutral mesons Roberts, Phys. Rev. D 96, 116015 (2017) https://arxiv.org/abs/1706.03378
It is a good thing that you do not have to distinguish between "good" journals and "bad" journals. It is a good thing because journals are not static objects. Good journals can become better and they can decay. Bad journals can improve. Good journals can publish bad articles and bad journals can publish good articles.
You should not judge people based on their commonalities such as skin color or religion. You should not judge articles based on their authors or the journal they are published in. You do not judge based on skin color no matter how good the statistics you have that prove how bad the average person of a particular color is. You judge people based on their merits and you judge articles based on their content. This is the way that science has been (correctly) done for thousands of years and you are not in a position to change that tradition. Truth is truth, no matter where it's published or who says it.
Yes. You should take a question on its merits. Ditto for an answer. If it's junk science you say so and explain why, and point out that the journal will print any old tosh for money. If it's serious sincere science you deal with it as normal, then you ask yourself why it appeared in a journal you don't rate.
The alternative is to close down serious sincere questions and/or delete serious sincere answers, which will drive contributors away. You should try to avoid this.
If no, how do we check if a journal is predatory?
You go and look up the journal and point out its submission rules. You don't just declare that it's predatory because it's appeared on some list. Note this from the Wikipedia article on Jeffrey Beall:
Joseph Esposito wrote in The Scholarly Kitchen that he had been following some of Beall's work with "growing unease" and that Beall's "broader critique (really an assault) of Gold OA and those who advocate it" had "crossed the line".
Also note that the word predatory is something of an ad-hominem, and that all journals are predatory to some extent. We fund science, then we've got to pay through the nose to read a paper that results from that funding. Not just a dollar or two, such that pricing is pro-rata comparable to the price of a book. But tens of dollars. So much that most people don't read it, and express their irritation at a paywall. Also note that journals that use this model do not like Open Access. They have an axe to grind.
You mentioned this question: What is the reason for the reported opposite-of-QM handedness of photons in a Bell test with circularly polarized photons? and a comment to this answer: https://physics.stackexchange.com/a/475637/36194. I took a look and thought it was all serious sincere science. So I think you should "tolerate" it. Especially since there are bona fide physicists who can't get their papers published in a high-impact journal. As to why, I'd say it's to do with the not mainstream issue. But that's one for another day.