Why it matters to protest, if somebody creates public nuisance

It is a common refrain "I don't want to poke my nose in somebody's personal matters" when somebody misbehaves in public.
But if matters are sorted in public, are they personal anymore!

Ignoring a public wrong amounts to encouraging hooliganism.With the legitimacy provided by mute spectators It becomes 'fair game' and spreads quickly. It is only a matter of time before you would find yourself target of one.

The past serves an example :
Martin Niemöller wrote about the inactivity of German intellectuals following the Nazi rise to power and the purging of their chosen targets, group after group.

First they came for the communists,
and I didn't speak out because I wasn't a communist.

Then they came for the socialists,
and I didn't speak out because I wasn't a socialist.

Then they came for the trade unionists,
and I didn't speak out because I wasn't a trade unionist.

Then they came for me,
and there was no one left to speak for me.