I'm not a hundred percent sure, but...I know in English it's still acceptable (Although outdated in my personal opinion :P) to use "man" to represent "humanity" or people in general, but I don't think it goes the same way for Portuguese, at least not in the singular form, as I've heard "Homens são..." used in a way to mean people in general.
It is a very old quote, the English saying "Man is responsible for his actions" is attributed to the 18th century philosopher Immanuel Kant. Armed with this piece of information I used Google to search for Kant along with the Portuguese sentence and my first hit (surprisingly from the Kennel Club of Brazil: http://www.kennelclub.com.br/obediencia/cao_cidada_02.htm) mentions this:
Immanuel Kant (1724-1804), filósofo da razão já ponderou: "O homem é responsável pelos seus atos e tem consciência do seu dever".
As this is a very old quote what you say could still be correct in modern Portuguese.
Kant wrote, "Der Mensch ist für seine Taten verantwortlich," with the definite article, required in German as in Portuguese. In English there is no exact equivalent. "Man" means "mankind," a collective. Der Mensch can mean "Mankind" but also "The human being," which is the literal sense here. "The man" distorts the sense of the dictum. Simply "Man is responsible . . " is possible, but sounds awkward.
So the Duo translation above is not accurate, if grammatically correct. The most natural way to say it in English would be "A man is responsible for his actions."
Philosophically correct English - the person - would probably be most true to the original Kant quotation. Fortunately, for the engineer in me, we are trying to learn Portuguese (and a lot of English that I did not learn in school).
Kant might have said: No easy answer to understanding people!
"... showing my age" No, no. It really sounds better, I've heard (or read) this phrase before.
To a native speaker this would definitely mean this man is repsonsible for someone else's actions, so it's an okay translation, but not of Kant's apothegm. We know the man is a "he." The "their" and "they" creep in when the antecedent is not gender-specific. In speech most people now would say something like "Every teacher is responsible for THEIR own syllabus." This is still offensive to my ears, and I would prefer, "Every teacher is responsible for his or her own syllabus." "If a teacher sees someone cheating, they have to report them" is colloquial English, but sounds awful. In writing we'd find some way to avoid the plural pronoun representing a singular noun. I hope.
"The man is responsible for her acts" is also good, but only if it means the man is responsible for what the woman did. Without the Kantian context, I think both have to be accepted because they do convey meaning in a grammatically correct way.
[I'm sorry, I first left out the word THEIR above and totally botched my meaning.]
I think I was in error above in allowing for a translation that makes a man responsible for someone else's actions. "Seu" has to refer to the subject of the clause. So "Men are responsible for their acts/actions" or "A man is resonsible for his acts." "The man" would mean a specific man, which I suppose is a possible intention here, but I take the sentence as a statement of a moral principle.