Why don't journal publishers ask for proof of affiliation?
The crucial point is: Why would the author lie? Let's try some hypothetical answers:
To bluff the editors and reviewers so that they think you're at a top place. But reviewers will likely be from your field of study and realize that you lie. They would probably know it if you moved to a high-ranking institution.
To make the paper look good in your CV. This is a non-sense, you sell your affiliation in different ways than by listing them in your papers.
I can't think of any other reason. Given that the authors have no incentive to list a false affiliation, there is no reason to verify it.
Also, remember that some affiliations are very hard to verify, for instance if you stay somewhere for 6 months and want to list it as an affiliation, you possibly do not appear in any official lists.
A journal is interested in whether the submitted article is,
- Appropriate in subject matter for that journal
- Of a suitable standard for that journal (as advised by reviewers)
In an ideal world, neither of these things is indicated by affiliation - so why should they care?
There are several answers here.
They quite probably already do so (eg if a claimed affiliation to a prestigious institution looks too good to be true, or unlikely given other information, or a reviewer says "hey, wait..."), but on an informal and ad-hoc basis, rather than doing it for the 99% of unremarkable cases.
They usually don't need to. Most submitters will provide an institutional email address, which is itself a fairly good indication that you are affiliated with that institution. (In the old days, you might have used letterhead - same sort of thing)
Beyond this, defining "credentials" would be complex. Would you need them to point to an institutional webpage with their name on? Submit a payslip? Produce a certificate of employment? (And what would you define as "counting" for affiliation?)
Finally (and most importantly) most of the publishing system is based on trust. The publisher trusts you to have actually carried out the experiments, and to have reported them honestly and comprehensively. They trust you not to have plagarised, or committed ethical breaches, or misrepresented other researchers. They may ask you to sign something to certify you've done all these things correctly, but they won't ask for evidence that someone else has verified you did them. If they're willing to take your word on the actual content of your science, why be particularly distrustful of your affiliation?