While a good analogy to a point, internet trolls do not have much of an impact on anybody unless you let them. Being the president gives a person pretty decent power over a ton of people. Not sure if ignoring the problem makes it any better.
I guess what I'm saying is be selective. If he's tweeting about Obama or Hillary or someone that has no impact on anything, why take the bait? If he's tweeting about taking credit for something he didn't do, ignore it. If he's tweeting about war with North Korea or immigration policy or something that might actually have an effect, then maybe pay attention.