Looking into safe haven laws in NJ, it seems like they only apply to infants 30 days or younger. So if he was 2-4 months old, the safe haven law would no longer apply and it sounds like the mom would be prosecuted regardless of where she left him.
Also, is there a considerable difference in care between a baby that age that has DS and a baby that doesn't? You would think there wouldn't be, since all 2-4 month olds need constant care no matter what. Therefore I doubt that he was abandoned because he was "too hard" to raise. If that was the case, I'd assume the mom would wait until the baby was older and the developmental problems presented with DS would become more noticeable. And if it was a stigma-related thing, she'd probably give him up for adoption at birth instead of wait a few months.
Maybe raising a baby in general proved too difficult, or she couldn't afford to properly care for him?
I don't want to judge the mother here, since we don't know the circumstances and we're not sure if the woman who left him with the babysitter was even his mother. I'm just thinking out loud.