Although minimizing harm is always brought up here and there, it seems to be overshadowed by the responsibility of the press to give the truth to the public. I'm not saying there's anything wrong with that, but I think there are definitely ways in which harm can be minimized without losing the truth of the story. Some media may be going for shock factor in their stories, but that factor can be what really causes harm to people.
As I write this, I have the publishing of pictures specifically in mind. In the SPJ Code of Ethics, it reads, "Be sensitive when seeking or using interviews or photographs of those affected by tragedy or grief." A lot of the time, the privacy of people, especially in tragic times, should outweigh the public's right to know or at least the right to know every last detail. However, it can be pretty hard to win a case if a person has been invaded of their privacy by the press. In the case of a public person, depending on the invasion of privacy, they may have to prove actual malice to win their case.
My hope is just that no matter the person, public or private, journalists still keep in mind how publication might affect a person before publishing. I'm definitely not saying to hold back information just because it hurts a person's reputation, but if a person's gone through a tragedy, he or she may not need to be reminded of it again if the public interest in the tragedy isn't huge.