I'd say that it's
always bad for people to be beaten once arrested. Or with violence for exercising their rights.
But I'd put the "optimal" (in the efficiency and good-social-outcomes sense, as distinct from a morally good-bad action perspective) amount at >0.
A credible threat of violence by police is a powerful deterrent. Especially to people who don't see the world in more nuanced light.
If someone like that is afraid of the police and so doesn't commit a crime, then everyone wins -- the potential perpetrator (who isn't exposed to the criminal justice system), the victim(s) (who don't have a crime committed against them), the police officer (who doesn't have to be in that situation and can focus on better parts of the job), and society as a whole (who save the substantial cost of dealing with all of the above: incarceration, rehabilitation, salary, lost productive economic activity).
The more nuanced discussion that tends to get lost in the weeds is "How do we simultaneously allow police latitude to do the job we want them to do, while also ensuring that latitude isn't disproportionately employed against minorities or political out-groups?"
In other words, police may need to crack a few hard heads. But those shouldn't be statistically-over-representative person of color heads.
Of course, fold in that economic situation is also racially biased, as a consequence of historical inequities, and actually accomplishing statistically-fair policing seems intractable. :(
But at the end of the day, people getting stabbed isn't good for anyone.
Even from an anti-police perspective, it builds support for "hard policing" policies among the general public, and eventually an election swings someone they support into office.
--
There was another comment in here that asserted that cops commit domestic violence more often than the general public.
Which immediately made me think "Why?"
Obviously, the acceptable amount of domestic violence is zero, but (if those statistics are true) there's something causing police officers to commit it more frequently than the general public.
It's probably partly that policing selects for "tough" people, by virtue of the work...
... but it also speaks to cumulative trauma caused by the job, and the effects of that.
You can't watch humans be shitty to each other, repeatedly, without it changing you. :(
It feels like we need a more complete lifecycle than "police for life," that includes cycling through various related components, some of which provide an opportunity for healing.
E.g. also stints helping defense and prosecuting attorneys, legal defense funds, partnering with mental health professionals, EMS, fire/rescue, etc.