Jesus actually had quite a lot to say about social responsibility. First and foremost, Jesus said that the
most important thing a person needs is to be saved. People need to have their sins forgiven and put saving
faith in Jesus Christ. This, more than anything else, will change a person's life. People who come to
know Christ have transformed hearts and minds. They desire to do good and to honor God
with their lives. They care about the way that they treat others.
But Jesus also gave some specific commands to Christians regarding social responsibility. Christians are called to care for the poor and lowly, to take care of the sick, to comfort the dying, to welcome outcasts. In the book of James, it says that "true religion" is caring for widows and orphans.
Perhaps the best way to phrase it is that Jesus calls us to care both for people's spiritual and physical needs. We care for their spiritual needs by telling about the free gift of salvation offered through Jesus. We care for their physical needs by serving them in meaningful ways; this loving service also helps them learn about their spiritual needs.