Christians believe that mankind is inherently sinful (original sin) and that the world suffers from the curse of sin in mankind as well.
It is true the evil will increase, but "where sin increases, graces increases much more." (Romans 5:20) The Bible informs us that before the end of the world there will be people from every tribe, nation, and language who will name Jesus as Lord (Revelation 7:9). So, even though things do get worse, at the same time the Gospel continues to spread to more and more people, and they experience the the forgiveness, joy, peace, and fulfillment that is only found in Christ.
Christians are absolutely genuine in their attempts to make the world a better place. It is instructive to see in John chapter 5 that Jesus heals a man who was an invalid for 38 years, and when people ask him who healed him, the man didn't even bother to get Jesus' name. Jesus healed the man not to get something out of him, but simply because He delighted in doing good.
As Jesus said elsewhere, it is better to give than to receive. Giving is its own reward, as is serving. There is true joy in giving to others and helping them in their times of need.
I would also say it most certainly is possible to make the world a better place. Jesus actually tells us that we are the "salt of the earth" (Matthew 5:13). Salt is a preservative. Christianity ought to hold up a very high standard of moral excellence to which unbelievers can aspire. Without that standard of morality, the bar is set much lower.
Also, historically speaking, as Christianity has spread, the world has indeed gotten better. Only when it decreases does the world get worse. Europe was full of barbaric people prior to Christianity. The gospel transformed it. Orphanages, hospitals, saving girls from forced prostitution, the feeding of the poor, the digging of wells... all this comes from Christian organizations all over the world today. In fact, it is a sobering thing to think of how dark the world would be without Christianity.