I don't believe a person has to balance their view of good things in the world with their view of the bad. The question shouldn't be "How can we balance being happy with so much suffering in the world?" but rather, "How can we suffer and still be happy?"
Some will wonder if we can, and the answer is that we can indeed. Philosophy aside, people do it every day. People suffer tragedies and hardships of varying degrees and yet they press on with great determination and continue to face life. A poor man may rise out of bed in the morning not knowing whether he will be able to eat dinner that night, but he still wills himself onward. Still, sometimes simply surviving is not enough. A life can feel very empty if it has no happiness.
Spending some time in Smolensk, Russia with orphaned children, I saw a system that had abandoned its people, which it was created to protect. Living in rundown boarding schools and camps, where crippled and sickly children sometimes must walk 100 yards from their living quarters to their shower building (imagine having to do this every day in the bitter Russian winter), they are regarded by some citizens and by the government as lesser people. Little is expected of them and little is given to them. When they turn eighteen-years-old they will receive papers release papers from the school for them to go out on their own and they will most likely either join the military, work in some menial job, or become involved in crime. A person in this situation has far fewer options than the average middle class person in the western world. Can a person under these circumstances be happy? Yes. Living with them for nearly a month, I can tell you firsthand that these children and teenagers were happy and full of life.
Of course all human beings suffer from time to time, but we must admit that some undergo fewer hardships than others. If a person who most would think is suffering a lot can be happy then I would hope that people who suffer less would be able to be happy.
If a person who is suffering is also happy then it would seem that happiness and suffering are not linked. And, of course, a person who is not suffering can also be unhappy. So what is it that causes happiness and unhappiness? Are there causes?
Certainly there are things that can cause temporary elation, anger, or sadness; but those emotions are just that: temporary. It is our will that controls our happiness. The philosopher of Stoicism known as Cleanthes said that the wicked man is "like a dog tied to a cart, and compelled to go wherever it goes." This is how you are when you are unhappy, living through unreason. On the other hand, Epictetus, who himself was once a slave during the time of the Roman empire, said that one could remain "sick and yet happy, in peril and yet happy, dying and yet happy, in exile and happy, in disgrace and happy." He also said, "If, therefore, any be unhappy, let him remember that he is unhappy by reason of himself alone."
I believe that it is possible to be happy and to suffer, but not everyone does so unfortunately. If one can manage to be happy this does not mean that one should be complacent with their suffering. A person would prefer not to suffer. And though there is suffering around you, it is possible to be happy. This does not mean ignoring the suffering around you. One must live his life and do what he can to stop suffering, allowing for position and ability.
As Marcus Aurelius said in his Meditations: "Say to yourself in the early morning: I shall meet today ungrateful, violent, treacherous, envious, uncharitable men. All of these things have come upon them through ignorance of real good and ill... I can neither be harmed by any of them, for no man will involve me in wrong, nor can I be angry with my kinsman or hate him; for we have come into the world to work together..."