Hey Phil,
I think the DRO's wouldn't have to deal with this sort of thing that much.
In the absence of the state, people would plan having children much more carefully. E.g. the Finnish state pays big bucks to people with children, and it makes me sick to my stomach when I see people making babies just to score at the Welfare Office. In the status quo, having a kid will increase your income, and you basically achieve immunity from fiscal responsibility.
If the state weren't there to reward these disgusting individuals, they would be horrified at the very prospect of having a child - which would be the proper reaction. The state warps (even removes) the cost/benefit analysis that every responsible person makes, especially when having children. Of course, the state wants more tax cattle, so they pay idiots for breeding.
The whole thing makes my blood boil. 
Ahem, anyway... 
If you remove the state from the equation, the inevitable result would be 1) less children and 2) more responsible parents.
Of course, the DRO's would be highly incentivized to encourage this sort of thing, but the demand for such services would be significantly lower when you consider the customer base in "the DRO world" and how they would think and behave.
If the DRO model were in place, I'd invest in companies that produce prophylactics and/or birth control products. They'd be making solid business.
If you would convince a man that he does wrong, do right. But do not
care to convince him. Men will believe what they see. Let them see.
- Henry David Thoreau