Along with a limit to how many children you can have, something like this might work without being considered eugenic. Allowing anyone to breed a few times, but not necessarily allowing everyone to raise kids.
Educational courses and basic psychological evaluation could be the requirements; but they would have to be very reasonable and have actual usefulness to even the competent would-be-parents. There should also be advanced education and rehabilitation for seemingly incompetent would-be-parents. Maybe supervised care should be the only common requirement for parents deemed incompetent. It should be a rarity for someone to not be allowed to raise kids on their own if they go through the system willingly.
A gradual increase in standards, starting very low and giving time for people to adjust as they rise, might ease any shock the system could have on the culture and be a good way to implement it.
All requirements should be based on conventional scientific knowledge, no strong idealism—even common idealism such as all children must have a reverence for the law. Children should be taught the law, and the consequences of breaking the law in their education, but the belief that the law should be revered is subjective and it shouldn’t be required of the parents to support it.
The vast majority of violent criminals or otherwise unproductive citizens come from bad homes. You can use crime-rate stats and how they correlate with the legalization of abortion in the US to illustrate the benefits of making sure citizens have healthy childhoods. (I’m sure there are other stats you could find that imply the same thing.) If childhoods could be systematically genuinely improved by the system, that would justify the cost. It would make more productive, less dangerous citizens.