I struggle with this too sometimes, from a couple angles. Primarily, I’m worried about how people perceive me. I’ve been practicing not giving a shit what other people think of me for about a decade now, and I’m pretty good, but it doesn’t always work. I have to remind myself that it just doesn’t matter if a stranger disagrees with me, or thinks I’m a fuckwad/idiot/etc, just like my opinion of them doesn’t change their life. We don’t know each other. Their negative opinion of me has no real impact on my life, and holding onto that helps me move past caring about their opinion.
Secondarily, I stress about misinformation/toxic ideas being spread. I pipe up in a lot of discussions about feminism and the patriarchy because I want to clarify misconceptions that a lot of people hold about these issues. But, as far as I know, I’ve never changed someone’s mind, and the effort just brings me down. I have to remind myself it’s not my responsibility to teach someone why they might be wrong. The odds that I would actually succeed are very low, unless the person is genuine and asking questions in good faith.
ETA: lastly, I ask myself how I want to feel today. Do I want this random asshole to ruin my day? I don’t want to give them that power over me. So I work to take back that power and make sure I have a good day in spite of their efforts.