In some cases the answer is obviously yes. Off-duty cops committing crimes, not good. A politician or judge might never be fully off-duty.
Where is the line for professors? Naked pics with their spouse? Orgies? Racist tweets? Racist comments at a friend’s barbecue? Being a spy for China?
What about for regular employees. Fired when they speak against the company? When they complain about toxic work environment? Or just if they say something racist? What if they’re asian and say a racist joke at a party? Or an ardent feminist with edgy views against men? How about an angry father who got his kids taken away and now says mean things about women?
If we normalize these firings for saying things we don’t like, others will want to normalize them for things we do like.
I think this is probably the median perspective.
For example, firing people for joining the KKK is a good way to reduce their ability to recruit and radicalize others.
But under American law you can be generally fired for no reason, so it's weird we are even having this conversation in the land of no job security.