Yes, his reputation is suffering because his wife wants to be an advocate for eliminating cyber bullying. He is one of the biggest bullies online that has ever existed. He attacks people for disliking him. In that way, he wants to be more like a dictator than a president. He cannot take challenges from anyone.
Trump is the President elect now. He needs to learn that he is above sending Tweets at 3 a.m. Trump needs to be dignified and he needs to learn that people are going to say all kinds of things about him. Holding a public office it is impossible to please everyone. He should chill.
Trump's Ongoing Twitter War: His reputation is not suffering. In fact, the more he tweets the more his popularity increases. Why? Because people have finally elected a leader who has a spine and is not afraid to challenge the media and fight back. I find it comical how the left reacts to it.
While it is possible that the ongoing Twitter war will hurt Trump in the long run, there is probably very little effect to his reputation at the moment. A lot of people are still riding the thrill of his win. However, if it keeps up, Trump's reputation might suffer later on. His appearance as president will reflect the way he carries himself on line.