Liz Williams has a great post about how TV can make one blasé about suffering.
This is why I don't watch the news on TV and have not done so for about 2 decades (deliberately so for the last decade). Watching things on TV inures us to them - it is external, somehow unreal, "just a story" — an attitude which is encouraged by the way the newscasters constantly refer to items as "stories".
I listen to things on the radio, where they frequently make me cry. This morning I heard a story about the death of a 17-year-old pilot in WW2, which made me cry. It was on Desert Island Discs with Martin Shaw. His mother met a young pilot in a tea-shop, just before he was sent to war. They danced to a song called J'attendrai by Tino Rossi. He asked if he could write to her. Three weeks later she received a letter from his commanding officer saying that he had been killed on his first sortie, and they had found her address in his wallet.
I frequently cry at things on the radio, because the pictures are in my head, not external to me on a screen. As someone once said "The pictures are better on the radio."
When they recited the list of those killed in the World Trade Centre, that made me cry. Another occasion when I cried at radio news was a story about some Australians crossing the desert to free some asylum seekers from a detention centre. It always makes me cry when people help others where there is no particular benefit to them for doing so.
Another way to prevent oneself becoming blasé about news is only to read the weekly newspapers, which summarise the events and give a slightly longer-term perspective. (This was an approach recommended by Thomas Merton.)
Of course, if one spent all one's time weeping over the suffering of others, one would be completely useless for anything else, including doing something to alleviate it — but I still wouldn't want to ever become inured to it. There is a sort of mental trick whereby one can set aside one's involvement and focus on solving the problem and seeing it objectively, but that takes some time to acquire.