Honestly, I think your initial answer did an excellent job of being unbiased. You used the terms of
accused of rather than
guilty of, and that's all that's really needed for your answer to be unbiased.
I don't know why your answer was downvoted(I up-voted it), but there are a few reasons I can Imagine as to why it might be downvoted, and they come from all sides of the isle.
note: pro-kasich and anti-kasich are localized to this specific statement
- A Pro-Kasich individual might not agree that he's promoting rape culture, and would therefore downvote you. I think this would be a wrong-headed reason to downvote, because there actually are people who consider things like what Kasich said to be an example of rape-culture, and what you said does not give a reasonable reader enough information to know whether or you not you agree with those people.
- An anti-Kasich individual feel like you're you're reducing all of rape-culture to somewhat reasonable-sounding comments like the comment that Kasich made.
- A neutral individual might feel like you're not going into enough depth as to why people feel that this specific statement is rape culture.
As for doing this in the general, I'd advise that you just keep your emotions in check, and use your common sense when writing your posts. If you feel passionate, and if you feel like you're fighting against an enemy, that's your bias you're feeling, so you should step back, cool down, make a good-faith effort to understand the issue the way your opponent does, and then try and write an objective answer.
Focus on being technically correct with your answer. If you're bringing your own analysis to your answer, think extra hard about how solid it is. Can you reduce it down to axioms that most people can agree with? if not, you're probably being biased. Changing your way of phrasing is mostly a cosmetic change, and if your answer was biased before you changed your phrasing, it will probably be biased after you change your phrasing too.