Clergy, by dint of our profession, often come onto the Internet to say things much as we would in a parish setting. But the Internet is not a parish. As such, we often find ourselves in situations without the proper orientation to move about successfully and fall headlong into the muck.
The Internet is not your home
In a church, you often know the people with whom you are speaking. There is a certain measure of trust you have for them and they have a respect for your office if not for you specifically. No one is paying the laity to be there so people who don't want to be at church simply aren't. Further, if someone is curious or concerned about what Father X really meant, they can walk over to him at coffee hour and ask.
No such dynamics are at play on the Internet. Online there should be no presumption that the people reading what you have to say have any affection for what you wrote or you personally. They didn't get up early to drive to your church, didn't stand patiently through your homily, and don't have to engage with you in the fellowship hall. They might not like you particularly much at all.
The Internet is all about sifting and sorting. Billions of things are being said every minute and so immediate decisions are made about the merits of anything you post. You aren't the single religious voice in a room full of hundreds of people. You are an almost random person saying something while 4.9 billion other people are doing the same. These are not the same activities and the gravitas of one venue does not in any way transfer to the other.
So, when someone says something rude about your blog post or tweet or YouTube video, the worst possible response is to fall back on your degrees or position or lofty titles. The Internet is a marketplace of ideas where sixteen year-olds garner millions of views and staid McLaughlin Group-types are so much detritus. Your ideas matter, your presentation matters, but your credits do not. Appealing to them looks weak and invites mockery (not sometimes, every time).
The question is not "Will someone say something unexpectedly rude for almost no apparent reason" but "How often will someone say something unexpectedly rude to me... today." A priest in his narthex that gets told off by an older woman for not showing up to pirohi making on time can kindly ask her to step into his office and talk it out. There is no remove to which a priest can go to resolve disputes online.
The alternative - arguing - is even worse. Imagine entering the boxing ring and not being able to throw a punch while your opponent tees off on you relentlessly. You as the priest are bound to things like decorum and forgiveness and rising above such bellicose pursuits. The netizen is free to pull your trunks down, kick you when you fall over, and take a photo with his phone.
Your best route is to calmly state your opinion, provide proof texts, and be polite to a fault. Ad hominem should always be avoided because there is no recovery from such a tactic. Finally, you need to remember that you can always walk away from a conversation. With a wife, family, parish, and all the attendant responsibilities, you have things to do. The person you are talking to might be on his second Monster drink and have no plans for the foreseeable future. Simple resource management should tell you that you are not equipped for a protracted argument with someone who quite probably has no interest in conceding to you on anything.
Who is going to clean up this mess?
What you do in a parish rarely spills over onto the Internet. But whatever you say online becomes the "hot goss[ip]" among your parishioners. It's not great when you "win" the argument, but it is so much worse when you very publicly lose. And losses online are screen captured and traded like Pokémon cards for decades.
A good rule of thumb is that more quickly you can respond to something, the less time you can take to think of the ramifications of your actions. A tweet can be sent off in seconds that will create such a stir that you will spend years cleaning up the mess. And, even if you can delete it, nothing is ever truly deleted from the Internet. Instantaneous communications should paradoxically be junctures for more time in reflection before posting, not less.
Trolls are self-employed
You have a wonderful job. Your wife and kids might occasionally make you late for church or spill something on your sermon notes, but you love them. You might have some cantankerous parishioners, but one day you'll be called to care for them in their illnesses. You have car notes, rectory expenses, and groceries to buy. You have a bishop who can call you at any time and redirect your life in any way he sees fit. The troll is beholden to no one. He's self-employed in his trollery. You can't fire a troll, but he can make your life miserable all the while getting even more "clout" and Internet adulation while he does so; a sword is the wrong tool against the Hydra of Lerna.
The Internet has brought thousands of young people to our doors. I have college-aged youths showing up at my parish every Sunday and you probably have them in yours too. It has been a blessing to us. In fact all of the people I will be bringing into the Church for Pascha found Orthodoxy online. The Internet is essential in our evangelical efforts, but it is not free from conflict. Conflict is in fact the primary draw for attention online so it is doubly important that you can engage online without being a casualty to it.
You don't have to like everyone
I will finish on this idea: some of the very people with whom you might have differences in the online Orthodox world are the exact same people bringing them to your door. Without naming names, we have rigorists on one side and progressives on the other that advocate for the Church but may do so in a way that you feel is unhealthy or out-of-balance. Speaking ill of the people who your inquirers probably very much admire - given that they listened enough to come see an Orthodox church for themselves - is not going to go over well. You can bring people to Orthodoxy without burning the bridges they are using to get to you. And, my goodness, are the trolls living under those bridges going to have something to say about you trying to burn down their homes.