Should I correct a mistake on an arXiv manuscript, that I found while refereeing it?
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.