I appreciate the feedback from everyone.
As several of you have correctly surmised, the scenario I outlined in the first post involves a relative I know quite well. I withheld many details (including gender, which hopefully explains much of the awkward phrasing) because I didn't want to bias the responses. I had hoped that people would play along more with the hypothetical and give some examples of situations that would lead them to report or not report.
To provide some context, this person is (IMO) nothing short of a toxic fiend with no redeeming qualities whatsoever: a self-serving free-loader who has repeatedly taken advantage of and caused lasting damage to many who tried to help. This person treats family (particularly those doling out the endless help) with savage cruelty, using the children as pawns and leverage to get what they want. The welfare and tax fraud bother me on principle, but the compelling reason to report would be to get the children out of the care of this person -- a horrendous parent, psychologically abusive for sure, likely physically abusive, and certainly negligent due to the drug and alcohol abuse.
I have not reported this person primarily because I'm concerned that others (especially the relatives who are the de-facto guardians of two of the children) will suffer some hardship, including legal consequences, as a result. The two children under the care of this person would almost certainly be better off in any other environment, even if they were put into the foster care system. And I'm saying that with eyes wide open; I'm aware of how awful the outcomes can be in foster care.
I admit I'm a little surprised that most of the responses were along the lines of "...it's none of my business/I might not have the facts so I would not report". Though I suppose if one assumed an incomplete understanding of the individual and/or situation, then one could imagine summoning sympathy for this person. For the record, states providing public assistance benefits are typically very interested in hearing about potential fraud and abuse.