They can't go after your children's accts. as long as they aren't jt. accts. with your name on them.
Since N.Y. is not a community property state, your wife is not responsible for your debts. They can't go after your wife's income/assets in her name only.
There is a "doctrine of necessaries" (that can apply even in non-community property states) that can make a spouse responsible for certain debts of the other spouse. However, in N.Y. this mainly applies to medical debts & then only in certain situations/circumstances.
Your wife & children's accts. should be ok. (safe) The normal procedure in a non-community property state is to go after your non-exempt income/assets & any non-exempt joint income/assets that have your name on them also.

