This doesn't seem to support any allegation that the NYPD "publicly threatened" DeBlasio's "children". What did happen is that the NYPD Sergeants' union inappropriately tweeted details about her arrest, after she was arrested. That's one child, for those keeping score, not "children", and it's also not a "threat" of any identifiable kind. It is definitely inappropriate, though.

So it would appear that Business Insider's headline was sensationalist and not well supported.

It was not a coincidence they arrested her when she did. It was absolutely a tactic to intimidate DeBlasio. It worked too!

>> inappropriately tweeted details about her arrest

This is typically considered a threat by most well adjusted people, and most definitely an extreme level of harassment.