Why do you put "her responsibility", instead of "his responsibility", as captain's grammatical gender is masculine?

English doesn't generally have gendered nouns, and the few it has are generally going out of use. I'm not aware of captain as being gendered in English.

I grew up speaking english, including reading a few older books, and I'm not aware of any gendered nouns at all. Which ones are you thinking of?

I was thinking of pairs such as steward/stewardess, actor/actress, and fiancé/fiancée. They're not gendered in the sense of nouns in French or German, but are gender specific.

How do you choose between his or her in this case?

This is more involved than I care to get into here. Here's a Wikipedia article which discusses it:

https://en.wikipedia.org/wiki/Third-person_pronoun

Why create an account to make your initial comment? It doesn't seem very substantial or on-topic. The two reasons I can think of aren't very charitable, which is why I'm asking, assuming that it's neither of those.

My charitable explanation is that it's a non native speaker of English genuinely asking the question to learn.

You don't, unless it's to create off topic arguments/trolling.

To answer your question if you know the sex, you say the sex.

If not you use sexless pronouns like "their" and "they" even if they are not 'proper' by old strict standards, basically they are, because language evolves.