When journals stop publishing them?
Depends on which journals. When CF was first announced in '89, major journals like Science and Nature published several papers on the subject, but when it devolved into an irreproducible phenomenon they quit publishing them. That is exactly what was to be expected though as Science and Nature are 'hot topic' publishers, and while a controversy is interesting, it isn't what 'hot topic' journals are looking for. (Yes, the editor of Nature made some poorly chosen remarks at the time, but that doesn't alter the fact that the controversy made them unfit subjects for Science and Nature.) Publication them moved to journals whose topical fields were consistent with the CF work, i.e. Fusion Technology (renamed later to Fusion Science and Tech.), J. of Electroanalytical Chemistry, Physics Letters, etc.). Then those journals started to cut back, primarily because there was no resolution to the conflict and lack of reproducibility occurring. So then the CFers started up their own publishing sub-network (Proc. of ICCFxx, J. Cond. Nuc. Mat. Sci., etc.). So technically, to date they haven't stopped being published, so I believe your criterion is not adequate. And besides, the CFers just claim 'establishment suppression' anyways.
Personally, I don't think you should ever stop listening completely, but when a field has persisted in producing non-reproducible results for many years, and when the prime researchers in the field clearly ignore critics and make gross errors in their response when they don't, others are at least justified in requiring some semblance of good scientific practice from the researchers before they are given any significant attention. That's just my opinion though. I was wondering what Epi thought was the criteria. Maybe I can be convinced I *can* ignore them...