If you are into eugenically improving the human species, that is exactly the opposite way to go about it. Being wealthy doesn't mean you are better. If anything, it means that you are more dishonest, cut throat and willing to kiss ass until your lips catch on fire. Also, look at Charles II. He was a freak with a jutting lower jaw. But by your standards, he would be genetically superior.
It has nothing to do with eugenics or genetics period. What I suggest here is the same thing that happens to working people.
Working people live on a fixed income. Because of that, some don't have any children at all. Others have a limit which is usually around 2 children. When they reach their limit on children they can afford, one or the other practices birth control. They have no more children.
From a liberal prospective, it's perfectly fine if taxpayers have to limit their family size because of income, but unfair if those taxpayers insist the people they support do the same.
Now if you want to talk genetics, yes, poor people generally have poor offspring. Middle-class generally have middle-class children. Same goes for upper-class and the wealthy. There are always exceptions to the rule, but it's the exception and not the norm.
Now if you want to stereotype all wealthy as dishonest and cut-throat, then all the poor are drug addicts and criminals. Which group would our society be better off with?
What you said had everything to do with eugenics. You talked about getting people fixed if they were on public assistance. It makes perfect sense if they are negro or latino. But getting White people fixed implies that there is something wrong with the children they would have. Most often, that wouldn't be the case. Also, long ago, the Prime Minister of England spoke about the "eccess poor" that the wealthy always complained about. He basically said that the Duke of Wellington was gald at the Battle of Waterloo that England had "excess poor." To have fighting for him of course.
Next, no couple would willingly have no children. Next, White people do tend to limit the number of children they have. But it has nothing to do with their income. Next, the children families on assistance are allowed to have should be limited. (Unless they're White) But getting them fixed so they can't have children at all, even if their financial circumstances improve, is going as bit far.
Next, of course poor people have poor children. It's like the "untouchable" class in India. Of course any children they have are going to be "untouchabe" too. But none of that has the slightest thing to do with genetics. At least very very very little. And the same goes for middle class and upper class children.
Next, nothing I said has anything at all to do with "stereotypes." It has to do with cold hard fact. First, the more people have, generally, the more they want. And I have never heard of a wealthy person who didn't want to become wealthier. It is also well known that the more wealthy somebody is, the more dishonest they are likely to be. Because consequences aren't something that comes to play in their lives as it does to a poor person.
Even status can promote dishonesty. I saw a study once where they would bring a child onto a room to take a short test. The answers of the test were put on a piece of paper face down on the desk. Which they were told not to look at. The children in this study were divided into two groups. Half of the children were given small desks to sit at and half were given large, full sized desks to sit at. The children who sat at the large desks were far more likely to take a peek at the answers.