Or how to create class warfare without really trying.
Welfare programs are kept strictly limited to the very poorest people for a reason. Once something becomes a universal program (like say social security or free public education for K-12) it is nearly impossible to kill. Conservatives know this. As long as a program is kept restricted to the very bottom of society, the middle class can see itself as benevolent when the mood strikes and will also be less concerned when those programs are de-funded or scrapped altogether.
Let's look at Social Security. Social Security is the third rail of politics- touch it and die. Bush only thought he could get privatization of Social Security though because he had a Republican House and Senate and a large number of younger Gen Xers that think that Social Security isn't going to be around for them anyways- why pay into it now. Even with that- privatizing Social Security was a no-go. It is a universal program, meaning everyone regardless of income gets some benefit from it. There is no means testing (though certain Republicans have floated that idea too) because everyone pays into it and even the richest of the rich get some benefit. Fair- absolutely. They may not need the money, but they paid into it too.
Now let's look at 2 programs that do basically the same thing- they provide people with no job a small income for a limited amount of time- Unemployment insurance and TANF (Temporary Aid for Needy Families). Unemployment is mostly not means tested. The amount of savings, investments, property that you own makes no difference in whether or not you'll receive Unemployment. If your spouse or children work- that makes no difference to Unemployment. If you receive child support- that makes no difference to Unemployment. You simply have to have been laid off from your job and have worked enough hours to have paid into the system. And you have to certify every week that you have looked for work. You don't even have to go into the unemployment office most of the time, just keep a log of where you applied and submit it when asked.
TANF on the other hand is means tested. You cannot own a newer car, have money in savings, a 401K, a retirement fund. You must be flat broke. No one in your home can have any sort of income. If you get child support- it will go to pay back your TANF grant before it goes to you. You must agree to participate in a job search program of no less than 35 hours a week. You must be in the Welfare office everyday looking for work and participating in really asinine programs to teach you how to be "professional". You must agree to having a whole hell of a lot of your life examined by a caseworker.
TANF has had it's programs slashed while the participants requirements have increased. Unemployment, not so much. We don't hear much about the Unemployment Kings taking our hard earned money and spending it on beer and hot rods, but we hear a whole hell of a lot about the worthless single moms on Welfare. Welfare is not universal, but unemployment is for anyone who has been laid off.
So passing SCHIP 10 years ago when it was just supposed to be for kids whose parents were too wealthy for Medicaid (most people who are not on Welfare, including those on Unemployment are too wealthy for Medicaid) but too poor to afford insurance on their own was easy. It's still keeping the means test pretty strict. But once you start including more middle class families, it starts to look more like a universal program. And universal programs generally grow (hello Medicare Drug plans) not shrink.
So when you hear a Republican crying about how SCHIP would cover "wealthy families" what you are really hearing is "and once those families realize it works better than our for profit system, we are never getting rid of it". It's much easier to take from the poor than it is to take from everyone.