Should I correct a mistake on an arXiv manuscript, that I found while refereeing it? I recently received a request to referee a manuscript for a journal and, while doing so, I pointed out a crucial mistake undermining most of the calculations. After two unsuccessful answers on the part of the authors (where they addressed other concerns of mine, but basically ignored the mistake I was pointing out), I recommended the journal reject the paper and they did so. 

Very recently, and within a couple of weeks after this rejection, the authors have openly posted their article on arXiv, without correcting their mistake. While I understand that manuscripts from arXiv should be taken with a grain of salt, as it is not peer reviewed, making a clearly incorrect result public seems wrong to me, both from an ethical standpoint and from a practical one as it might mislead researchers.

I am unsure how to proceed. Should I ignore it or take some measures such as publishing a reply/correction? If the latter, is there any way to do it for arXiv submissions? Is it possible to do it anonymously?
I think it is commendable of you to care, but it would be best for you to keep anonymity with respect to the authors.

A good middle ground would be writing someone who knows your identity already, and is known by the authors: for example the editor in charge of the paper for the journal you reviewed for. Typically this editor is known to the authors, and they could pass on your comments without disclosing your identity.

In any case, the editor should be able to at least give you a qualified opinion on how to proceed.