My wife and I sponsor a child each through a group called
http://www.compassion.com.au/
We are so privileged in our country, it's a tragedy that we have so much and share so little.
I have been to a compassion area in El Salvador and seen the difference it makes not just to the child, but their family and their town. I know others that have been to the compassion areas in (I think) Malaysia.
These groups do make a difference in peoples lives. It is my view that the more money you have, the greater responsibility (And ease I might add) you have to help someone else.
As many have mentioned previously, if nothing else, it provides education and education comes hand in hand with health.
The simplest of things like having a tooth brush and known how to use it makes a huge difference to health.
One place I went in El Salvador, they had to walk 7 km each way with their bucket, and climb a hill that we here would consider a 'scramble' if you were hiking - it's hard going without a bucket full of water and kids in tow.
I appreciate the sceptical view of people, I still have that view at times. But regardless of how much it costs to get it to them, it does get there, but of course there is overhead involved.
I do know people that work for this kind of organisation, and I also know they get paid less than they would at a mainstream organisation - that is to keep costs down.
Sadly it does not always work the way you hoped. My sponsor child in El Salvador fell in with the local gang and was removed from being a sponsor child. Knowing how violent the gangs are in El Salvador (look up MS13) it broke my heart. I know have a sponsor child in India.
Some other links for those interested.
http://www.makepovertyhistory.com.au/
https://www.oxfam.org.au/
https://www.oxfam.org.au/explore/millennium-development-goals/