Is it our responsibility as Christians to convert the world to Christianity? The great commission Jesus left us—“Go into all the world and proclaim the gospel to the whole creation.” (Mark 16:15 ESV)—would seem to indicate that we must.
The problem is there are plenty of examples of Christians whose efforts to spread the gospel are an unmitigated disaster—televangelists who use the gospel to enjoy a lavish lifestyle, priests who use their position to prey on young children. And then just look at all the different Christian religions and all the different denominations within the Protestant religion and their differing views of salvation. If I was an unbeliever, who should I believe? The problem is that we Christians are sinful and finite creatures and we do not always correctly present, in word and deed, God’s plan for our salvation. So why would God use such a defective means to communicate a message on such a critical subject—the eternal salvation of our souls?
The Bible tells us the Holy Spirit’s responsibility is to convict the world of sin, righteousness, and judgment (John 16:8) and he can do a much better job of that then we can. Our job is to assist in that effort by being a light on a hill (Matthew 5:14-17), by being the salt of the earth (Matthew 5:13). I like the way Rosaria Champagne Butterfield states it: Salvation is God’s work; service is what our responsibility is—selfless love and sacrifice. 
As for the great commission, my dad has said for many years that a better translation of it is: As we go about the world we are to share with others what Christ has done for us. Our responsibility is not to convert the world but to be an example of the impact Christ can make on the life of one person. God will take care of the rest.
 Rosaria Champagne Butterfield, The Secret Thoughts of an Unlikely Convert, Pittsburg, PA: Crown & Covenant Publications, 2012. P. 67.