... I have allergies, I have asthma. Have had it since I was a child.
I have had pets, all my life. Pets with fur etc. indoors or outdoor pets.
Now, as such, I know how it is, when your HEALTH... is at risk, and THUS, the Doctor recommends NO pets especially in the house. Because of your condition. Keeping a pet in the basement, does not... contain the allergens/dander in the house. It does migrate elsewhere in the house.
Anyway, my conditions are not as extreme as yours. Hence, I can have pets. And my Asthma does not have the same triggers as yours.
HOWEVER, it is VERY VERY sad... when other family members, seemingly think that their beloved pets, is more important than their Wife's or Mommy's, health. And then it puts a guilt trip on you and then it is YOU... who has to deal with it, because it is "your" problem.
But I think, that if you got very sick, because of this, they would be very regretful, about taking the pets over your health.
My Mom, developed Asthma later in life. And it was exacerbated by the pets we had. Indoors. The Doc told her we had to get rid of the cat, etc. But well the cat died. He was old. But we could not get anymore indoor, pets, with fur. It just was. And we didn't make her feel guilty about it. This is her health. One day, my Mom got REAL sick... and had an Asthma attack, emergency was called and everything.
So at some point a choice has to be made.
And that is, what or who is more important?
Your health.
Or the pets?
But your Hubby's cat is old. Sick. And in the basement.
You/he should get an air purifier, FOR that room, that the cat is in.
And get one for the other rooms too.
It can help a lot.
And this can be an attempt to help your conditions, while for now, keeping the pets. Perhaps.
ETA: also, when people handle the cat(s), and then they come near you... THAT WILL AFFECT your allergies. Because, the animal's dander/fur, is on them and their clothing and hands. That is what my Doc and my Mom's Doc, has said also. Anyone who plays with the cat, needs to change clothes and wash their hands.
AND... any allergens/pollen/fur/dander, can also exist on people's clothing and hands, even if you do not see it, because there are allergen particles that are SO small, it cannot be seen by the naked eye. This goes for also, pollen etc. that you get on your clothes or skin, from being outside. So my Mom's Doc, ALWAYS recommended that a person CHANGE clothes and wash their hands or face, after touching a pet and/or after coming in the house from outside.