Actually, we are all kinda 'saved at birth'. God doesn't generally hold us accountable on something until we decide we're smart enough to form a position on it. He says we are born innocent (really, a 2 month old has no idea what's going on at that Baptismal) and we remain that way until we think we know enough to decide on Him. Problem is, we live in a fallen world whose ruling interests include making sure we decide against God as early as possible. So we are in some way born saved, it's just up to us to make sure we stay that way by accepting Jesus's sacrafice after being made aware of it.
For a peson to be saved they have to have the mental understanding of sin. Clearly, a baby or a young child does not have that mental capacity. While they go to Heaven if they die as a baby they are still not "saved". They are not held accountable, and of course if they are a baby then they have not sinned. I mean... they're a baby! There would be no need for them to be saved.