Frankly, this is going to progressively become a less critical issue since as time goes on there are fewer and fewer international routes remaining with international first class service.
I support a policy where you need to list the infant as such on your reservation and have to check a box acknowledging that you may be asked to take the baby back to coach (or even Business Class) if they get noisy and calm them down there.
However, if such a policy were put in place it would be detrimental to the airlines bottom line since they would have to either refund the difference in fare to those who get downgraded or because a potential customer would go elsewhere, because of their baby.
The other problem with this plan is that where do you suppose they will find the space in Business or Economy class where the baby would be taken to? Keep some seats open and unsold for the eventuality?
So my fearless prediction is that this won't happen. Before such, a "baby free cabin" might happen, but anything that requires refunding fares or leaving a few seats unsold in the back and running an empty seats is not going to happen. And no one will dream of downgrading due to a crying baby without refunding fare either. Either way it will be a PR disaster for the airline in question, even though some fat cats might like it a lot.
Frankly the problem is solved much more cheaply by giving each first class passenger a noise canceling headphone