You're mixing things here. It's like you say you didn't steal the wallet, and also there wasn't anything in it.
Either those children working is a bad thing, or it's not. If they're making money necessary to feed themselves and their families, then you should be proud of paying part of those money.
Of course, we don't really know if it was a good thing. We don't know if they were 17 or 10, whether they really needed the money or how many hours they worked. The situation is much murkier then the knee jerk "I can't exploit it". And unfortunately not trying to think more about it doesn't help those kids one bit.
"That someone exists hanging off a cliff is a bad thing. So I shall remove the cliff." The only way you can achieve a positive benefit for these children is by paying their wages. Removing that payment is removing the benefit, and nothing more. Nothing good will come of it, save for a smug look on your face as they go back home empty-handed for the day, or as the factory switches to making parts for Dell or Toshiba instead.
When I was 13 or 14, I worked picking strawberries in the summer in Oregon. It involved getting up very early, riding a school bus for a couple hours to the fields, working all day doing back straining work for little pay, and then taking the bus home. I didn't need to feed my family, but I wanted to work because I wanted to have some spending money. I'm glad I live in a country where that opportunity was made available to me. Is this "OK"? Were those "unethically" picked strawberries?
Either those children working is a bad thing, or it's not. If they're making money necessary to feed themselves and their families, then you should be proud of paying part of those money.
Of course, we don't really know if it was a good thing. We don't know if they were 17 or 10, whether they really needed the money or how many hours they worked. The situation is much murkier then the knee jerk "I can't exploit it". And unfortunately not trying to think more about it doesn't help those kids one bit.