"It is a good rule in life never to apologize. The right sort of people do not want apologies, and the wrong sort take a mean advantage of them."
-- P. G. Wodehouse, The Man Upstairs (1914)
... There is some truth to this, for sure. And it is certainly possible to be too apologetic. And if one finds one needs to apologize really frequently, one has better look at what behavior patterns one needs to amend!
But: I find that an apology can be really handy in two conditions: One is when one honestly has f***ed up. Own up to it, and apologize. It really makes a difference to the person one has wronged.
And another is when one is locked in conflict with another person, and there is a lot of unclarity as to who is right and wrong. This can be a complex picture, and can further be obfuscated by high emotions. And I find occasionally that if you press on, you may ruin a friendship. So it can be better to just swallow your pride and think to yourself: I may be right here, but I am not sure, and ultimately it is not important, and just say: "You are right, I am sorry."
I admit none of these things are easy. We all have pride.
And of course it is important not to let people walk all over you.
Some things take a lot of judgement.