When the parents clearly are unfit (drug abusers, domestic violence) take the kids and give them to adults who actually can care for them.
When the parents are crippled, you can provide supplemental assistance until the kids get big enough to help around the house. You also can use the boarding school/weekend visit model, or daytime/nighttime parenting.
When there is only one parent, of questionable competence, so that you do not have a clear case to take the kids, the best option is to pool those parents together, and assign the best of them (at least two, if there is one competent parent in this bunch you do not need more) to take care of the kids, since this is the highest responsibility post. The others should be put to work.
When there is only one parent, who is more or less competent (e.g. holding down a full-time job or some collection of part-time jobs at various points), the best option is to get another parent in the household. This can be done via marriage or co-parenting with other single parents (same sex or opposite sex). The boarding school and day/night parent options also are available, but clearly are suboptimal.
When there are two more or less able parents, who are working but can’t make enough money, the best answer varies, but usually moving to a different region with more affordable housing costs, re-training one parent to higher skill level, or shifting more of the load to one parent (e.g. 60 hour work week) and then having the other parent stay at home.
What are the costs and means associated with these different options?
Adoption/foster care is the most expensive by far, since you have to employ able adults, which implies cost of ~$30->$50K/year if compensated at market rates.
Boarding school also can be fairly expensive, depending on whether you use an existing-house model, or a separate boarding facility. Day/night parenting shifts also work, but you’re still looking at ~$20K a year, assuming you can get that fractional allocation out of the available workers in the economy (and you won’t if you have a restrictive immigration policy, and/or old age pensions).
Pooling the parents together, if you have 4 or 5 in a house, is essentially cost-free to the state, except for the coercion required to form the unit against their will, and the cost to sort out domestic disputes. If these are sort-of-stable women, it can be fine – but sort-of-stable men can be a lot of trouble.
In the long run, shifting parental responsibilities is more or less neutral, but in the short run you usually are looking at ~$20K of retraining, re-housing, and similar transition costs in a well-coordinated system where you can better allocate the working parent.
Co-parenting can actually be negative cost, if the two (or more) parents like each other and work well together. It can require some procedural work because you have to work out roommate/rental agreements and so on.
Marriage also can be a negative cost, but the real issue is more about how it is structured, and how you entice a partner to marry someone who already has kids and is struggling. This particularly is true when the woman is too old to have kids, or can’t/won’t have more kids for whatever reason (and you would not want to force a woman to have more kids). The arrangement at absolute minimum has to be a good gamble; it may not be risk-free, but it should be clearly better on average, with a low chance of serious loss (< 10%) for the non-kid partner. Practically speaking, that means there will be no significant amount of alimony or child support coming out of the new marriage (although because there may be new kids, there still will have to be co-parenting after a divorce). Hence, depending on the treatment of a first marriage without children (a separate topic), you could consider marrying a single mother as essentially operating under a separate family law regime vs. marrying a woman without children – this discrepancy brings any number of other issues to the table that should be addressed. Because of the cost delta associated with a live-in parent vs. the boarding school or day/night parent costs, it also is rational (disregarding moral hazard) to provide a supplement (say $10K/year) for such an arrangement to pay for the labor costs in this case, for younger children up until about the age of 10. However, because of the significant financial benefit associated with this, it is an easily exploitable system, so I would not recommend offering such an incentive.
Of course one asks the question about where is the father, but in the general case the father’s economic or family utility cannot be relied upon. It doesn’t mean you don’t go after him for child support and force him to work, but if he’s rotting in a jail cell you can’t use that as a means of recouping childcare expenses.
The other question that comes up is about rich single parents – but although this clearly is a suboptimal case, it’s not a matter that requires the state intervention for basic needs. Consequently although you could apply any of the above approaches, you don’t have a compelling case to use force to apply one of them, especially since that single parent usually also has heavy involvement in the childrearing, so there isn’t the learning and socialization deficit that you usually are concerned about with the poor single mothers.