Ask, then do
Ask the family what would help this week, then do that thing. A specific offer ("I will bring dinner Thursday") is easier to accept than "let me know if you need anything." Nobody in a crisis has the energy to manage your good intentions.
Cover the basics
Set up a meal schedule for the first month. Buy diapers, wipes, and formula in the sizes they actually use. If rent or a car repair is the crisis, take up an offering. Men: ask the father what he needs, and fix what you can.
Give the father a man to call
A young father is far more likely to stay when another man tells him he can do this and shows him how. Pair him with someone who will pick up the phone at 11 p.m. and take him to breakfast on Saturday.
Stay for the long haul
Show up when you said you would. Check in after the first week, the first month, and the first birthday. Keep one point of contact so the family is not managing ten. Share nothing about them without their permission.