You are (and perhaps the other person also is) mixing up three questions.
1. Does doing X have costs? 2. If so, and if X is done, who should pay those costs? 3. Should we make X illegal?
In the present case, first-cousin marriage does have costs, one of which is that it increases the number of children born with genetic problems that make their lives worse and (if the government pays for medical treatment to mitigate them) costs money. So the answer to the first question is yes, and so far as I can tell no one disagrees.
If some pairs of first cousins marry, and their children have genetic problems, then some of the costs are paid by those children. In a country with socialized healthcare like the UK, the default is that some other costs -- those of helping the children to have longer and/or better lives -- are paid by the government, and hence by everyone's taxes.
You might want to argue that those costs are the fault of the particular people who entered into those marriages and they should have to pay them. Unfortunately, in many cases they probably can't afford to do that. You might want to identify some community that they belong to and somehow make them pay, but that's pretty difficult to do; the UK isn't set up to levy different taxes on (say) Muslims or people whose ancestors lived in Bangladesh or whatever, and on the whole that seems obviously a good thing.
You might want to argue that in any case the nation shouldn't be paying, and that the NHS should refuse to treat whatever problems those children might have. In that case it will be the children who suffer; I personally am not a fan of making children suffer for their parents' bad decisions, but some people are.
So the second question is certainly one that we could argue about. But the blogpost we are discussing never touches on that question.
What the blogpost is about is the third question. Answering that is a matter of weighing the costs of X against the costs of making X illegal (loss of personal freedoms every time anything is made illegal; making some communities feel persecuted; loss of whatever benefits, if any, communities in which first-cousin marriage is practised get from it; making X happen illegally rather than legally, as opposed to making X not happen).
That again is something that could reasonably be argued about, and the blogpost does argue about it. But, I repeat, it's an entirely separate question from the second one.
So I continue to assert that the article wasn't about whether "they should pay for the care of the people that do". It didn't discuss that question at all. It discussed the entirely separate question of what measures if any should be taken to make there be fewer "people that do".
> You might want to argue that those costs are the fault of the particular people who entered into those marriages
Yes, hence wanting to ban incest marriages.
> and they should have to pay them. Unfortunately, in many cases they probably can't afford to do that.
No, I and the other commenter are not arguing that. Make incest marriages illegal in the UK, make incest married people ineligible to migrate to the UK. It's very simple. This thread is begininning to resemble the "left wing discourse" quote so I'll stop here.
