What do do? Give the friend space and respect his/her wishes for privacy. “Repeatedly asking”??? That’s your problem right there. People can get annoyed if you keep something up.
Everyone handles their problems differently. For you, talking to people about issues might bring some comfort and relief. Others…prefer to handle things on their own and keep their issues more private. No matter which way you go about it…the important thing is for the person to find relief. And if your constant pestering them is adding to their stress as opposed to providing relief…then back off.
You can let the person know that you are worried about them and that you’re there for them if they need to talk. And then you just leave em alone and go about your life. If they need to talk…about their problems…or anything at all…they’ll approach you. But frequent asking/pestering can even come across as nosiness. Too much of anything…even a good things such as concern…can backfire. Give him/her space. The answer to your question is simple and I think you already know it.
You create balance by fulfilling the formality of asking if the person is okay and if they need help…and if they tell you they want to be left alone…you let them know that they can reach you if needed…and then you give space. The balance includes both the polite formality and respect for the individual’s wishes. When the individual’s wishes are not respected…that’s when the balance is off.