당신 (dangsin) is the word textbooks give for a polite “you.” Then learners use it, and Koreans wince. That is not because the textbook was wrong. 당신 is a real word and it does mean you. It just turns out to work in almost none of the situations where a learner would reach for it.
The short answer to how to say “you” politely in Korean is: you mostly do not. You use the person’s name or title instead, or you leave the subject out entirely.
Where 당신 actually works
Between spouses. Married couples, especially older ones, use 당신 for each other. 당신 밥 먹었어? (did you eat?) between a husband and wife is warm and ordinary. 여보 (yeobo) is the term of address, and 당신 is the pronoun that goes with it.
In writing addressed to no one in particular. Ads, song lyrics, poems, book titles, and slogans use 당신 to speak to the reader directly. 당신의 하루를 응원합니다 (we support your day) on a poster is perfectly natural. Nobody feels addressed as a specific person, so nothing goes wrong.
In an argument. 당신이 뭔데? (who do you think you are?) 당신 지금 뭐라고 했어? (what did you just say?) Used toward a stranger or someone you are not married to, 당신 sounds confrontational. It takes a step back from politeness while still technically being polite, and it signals that the gloves are off.
That third use is the one learners walk into. If you say 당신 to a shopkeeper or a colleague, it can sound like the opening of a fight.
The use almost nobody teaches
당신 has one more job, and it is the most respectful use of all. It can refer back to an elder who is being talked about in the third person, as a respectful way of saying “himself” or “herself.”
할아버지께서는 당신의 물건을 아끼셨다 (harabeojikkeseoneun dangsinui mulgeoneul akkisyeotda) = Grandfather took good care of his own belongings
Here 당신 is not “you” at all. It refers back to the grandfather, with honorific respect. This is mostly a written or formal-speech use, and you will see it in essays, biographies, and memorial speeches.
So the same word ranges from tender (between spouses) to hostile (in a fight) to deeply respectful (about an elder in the third person). That range is exactly why it is so hard to use.
So what do people actually say?
The person’s name or title. 민수 씨 내일 시간 있어요? (Minsu-ssi, are you free tomorrow?) uses the name where English would use “you.” With seniors, it is the title: 팀장님은 어떻게 생각하세요? (what do you think, manager?)
Nothing at all. Korean drops the subject whenever it is clear. 내일 시간 있으세요? is already “are you free tomorrow?” No pronoun needed. This is by far the most common solution.
그쪽 (geujjok). Literally “that side.” A semi-polite you for strangers when you do not know their name. It is usable but has a slightly awkward, distant feel, and it can also be used defensively in an argument.
너 (neo). The casual you, for close friends, younger people you are on 반말 terms with, and children. In speech it is often pronounced 니 (ni).
자기 (jagi). Between couples, often unmarried ones, as a pet name and pronoun.
Why Korean works this way
English gets away with one “you” for everyone because English does not grammatically mark the relationship between speaker and listener. Korean does, constantly, through speech levels, titles, and honorific verbs. A neutral polite “you” does not fit in that system, because there is no neutral relationship. Every “you” would have to say something about rank, and that is exactly the job the name or title is already doing.
The practical rule
If you are about to say 당신 to someone you are not married to, stop. Use their name with 씨 or 님, their title, or drop the word entirely. The sentence will almost always be more natural without it.