"The servant of the Lord must not quarrel, but must be gentle toward all people, able to teach, patient, in gentleness instructing those in opposition. Perhaps God will grant them repentance to know the truth, and they may escape from the snare of the devil, after being captured by him to do his will" (2 Tim. 2:24-26).
I do a lot of commenting on internet articles these days, and have found that what Paul says in this passage of Scripture is just as relevant today on the internet as it was 2,000 years ago.
The "servant of the Lord" presupposes the fruit of the Spirit in our lives (Gal. 5:22-23). Without this, the remainder is difficult to do.
Striving or quarreling is a good way to end any meaningful exchange between people. If someone responds in a negative way to one of my comments on the internet, and I care to respond to it, I don't respond with the same tone that they used toward me. Being "gentle unto all" is the way to not escalate the disagreement into an exchange of insults; instead, you go into the mode of "apt to teach, patient, in meekness instructing those that oppose themselves." If the person responds, you will usually notice that their tone is not as confrontational.
I always keep in mind that the purpose for the exchange is to teach. I also assume that there are others that are reading the exchange, and that my comments are also for the wider audience. If you have anything meaningful to say, you want to keep the exchange going until you have said everything that you intended. Many times the response that you get can move the conversation in a direction that you didn't anticipate at the beginning. This can also be useful.
An important point to remember is that we don't control the outcome of the conversation. Once it ends, God does the rest. God is the one who "peradventure will give them repentance to the acknowledging of the truth; And that they may recover themselves out of the snare of the devil, who are taken captive by him at his will." When we remember this, it takes the pressure off of us. There is no need to feel guilty or angry if the other person does not appear to agree with us. God may have to work on them as only He knows how. The anonymity of the internet prevents us from knowing the full impact of our words most of the time. This comes with the turf. Our job is to put out the Word. God disposes of it as He wishes.
"For as the rain comes down, and the snow from heaven, and do not return there but water the earth
and make it bring forth and bud that it may give seed to the sower and bread to the eater, so shall My word be that goes forth from My mouth; it shall not return to Me void, but it shall accomplish that which I please, and it shall prosper in the thing for which I sent it. For you shall go out with joy, and be led out with peace; the mountains and the hills shall break forth into singing before you, and all the trees of the field shall clap their hands" (Is. 55:10-12). We are to "go out with joy, and be led forth with peace." We can't do this if we assume all of the burden for the outcome.
Nature rejoices when we spread the word, "that the creation itself also will be set free from its slavery to corruption into the glorious freedom of the children of God. We know that the whole creation groans and travails in pain together until now" (Rom. 8:21-22).