There does not seem to be any information about the start of this myth in any official sources that I've been able to find.
Some probable reasons could be:
1. Children really are toxic - when they're sick with human diseases.
Perhaps, at some point in history, probably when measles, mumps, poxes and scarlet fevers were a serious danger, some Scarers brought diseases back from across the door. The monsters clearly have very limited understanding of humans - perhaps they 'discovered' that the 'common cause' in all those cases was that the Scarers touched kids or their things, and those flying Scarers who didn't touch anything got "poisoned" less, and came to the seemingly logical conclusion that it's children themselves that are toxic. And the myth perpetuated to the 'present' hysteria causing levels.
2. CDA perpetuates the myth to maintain the separation between the human and the monster world
The movie shows us that the monsters are not only kind of clueless when it comes to humans, despite the fact they've been in deliberate contact with that world for centuries at least, they are probably also technologically not nearly as advanced - the only way to generate energy they've been able to find in their civilization is from human kids! And then their entire economy seems based on getting that super-rare resource.
We also see what happens when a unscrupulous businessman realizes kids aren't actually toxic - he feels no qualm in kidnapping to get that rare resource.
Since the CDA seems to know more about all the stuff than other monsters in the film, it is not inconceivable that they may also be aware of the true destructive potential of humans, and the kind of wrath they could visit on the monsters if they discovered that not only were they real, they'd been kidnapping and torturing human kids.
So what's the best way to keep human-monster contact minimal? Make that contact have consequences too scary to even contemplate - note no one really seems to know how kids are toxic or dangerous, just that they are and everyone imagines their own worst scenarios - stimulated fear of the unknown at its best.
Mixing the two makes it even more probable - Maybe the myth initially spread due reason #1 centuries ago, but then CDA carefully inculcated that fear for reason #2.
The main problem I see with the often suggested fan-idea that the reason for the propaganda was so the monsters wouldn't feel bad about scaring little kids is there would be much more efficient ways to achieve that.
The rumor-mongering authorities could de-"humanize" the humans - they obviously can't even walk or talk properly and they look weird - they're just animals! Scaring them is really necessary for the growth of those interesting creatures, and the screams are actually screams of joy! Or just make the power extraction business classified so that the general public doesn't even know it comes from human kids. Or even what human kids are.
In the end of the movie, they switch to making kids laugh primarily because laughter is 10x more powerful, not because 'scaring those kids would be bad'. Even with all the experience with Boo, if SCREAM really was the only way for monsters to get energy, I can't see that entire civilization saying 'screw it, let's go back to the Stone Ages, because that's better than having to make little non-toxic kids scream a little!'
In real life we can be quite complacent about inconveniences suffered by distant others (kids or no) when it comes to getting our own creature comforts; I can't see it being significantly different for the monsters.