I assume you mean "stay at home" and not "work at home". There is a huge difference. The rest of this post will assume "stay at home" homemaker watching the kids.
First of all, if the dad is staying home (to watch the kids, rather than to work) doesn't it usually imply the wife is out working? Usually it's for a practical reason: she can easily earn more money than her husband (since the modern education system, K thru college, is tailored to the female mind. So the woman gets better grades, is more likely to go to college as a result, etc.)
Welfare is never so complete as to pay ALL your bills 100%. Depending on the state you live in and how much you earn, you might get food stamps, free healthcare, maybe a free phone -- but you're usually on your own for toiletries, non-prescription drugstore products (Nyquil, vicks vapo rub, aspirin, hydrogen peroxide, etc.), clothing, household goods, gasoline, utilities, house/car maintenance, furniture, car(s), toys, TV, any cable/Netflix/Internet subscriptions or entertainment, etc. I have some relatives in Illinois (a "blue state") so I would know! If they don't give it to you in Illinois, you probably can't get it anywhere.
So unless a close family member is subsidizing the family so that BOTH parents can stay at home, living off welfare in abject poverty, one of the couple has to go out and work. Having the wife do the working outside the home is backwards and most disordered.
Any "man" who considers (much less agrees to) such an arrangement shouldn't be considered a man at all. So his wife will go out and interface with the world at large, earning a living, while he stays home and concerns himself with domestic affairs? Insanity! If God wanted him to fill that role, He would have given "him" a uterus so he could bear children. The one who stays home and raises the children should also be the one (mother) who bears them. We're talking about nature here.
What happens when the woman gets pregnant? They both stay home and starve to death and end up on the street? Or do they just use birth control with "economic hardship" as an excuse?
Either way, I'd say: man up, pull yourself up by something or other, and make something of yourself so you can support your family! It's a grave obligation for every father.
And if you're having *any* difficulties along these lines, make sure you start by selling any computer or video games and/or systems -- and any other immature "toys" unworthy of a grown man -- if you have them. That includes boats, motorcycles, DVD collections, comic books, collectibles, and any other expensive hobbies. You can use the time you save to build up a real career so you can support your family properly!
In fact, I bet most men who want to stay home (while their wife goes out and earns "good money") probably have a well equipped "man cave" with many toys of this nature. They are happy to surrender their manhood/dignity in this manner, because they are immature overgrown boys to begin with.
PS --
God imposed one punishment on Adam (he would have to work)
God imposed two punishments on Eve (she would be under her husband's authority, and would have pain in childbirth)
We all castigate and look down upon (and rightly so) woman who are feminist, headstrong, who won't submit to male authority.
We all castigate and look down upon (and rightly so) women who use birth control and/or abortion to have zero children despite having plenty of "relationships" with men.
So if we chastise women for casting off the punishments imposed on them by God, why shouldn't we castigate men as well who similarly refuse to bear the punishment imposed on them by God?
"He who will not work, neither let him eat." -St. Paul