Asbestosis is NOT an infection - it's a mineral.

It is only a problem if you start ripping out large amounts of it without protective gear and breathe large quantities of dust. If just living in a 1960s house were dangerous, half of Phoenix would be dead by now.

The two risk factors for cancer from asbestos were:
1 - Heavy smoking
2 - Working in asbestos mining, manufacturing, or application ... without protective gear.

Asbestos is only dangerous when it is being disturbed and creating dust. You are really worrying about it for no reason.

Being a worrier and not going into the basement is not a reasonable approach. Either forget about asbestos or go into the basement and look for it.

Even if there is asbestos in the basement, unless it's disturbed and flaking off into the air (which is doubtful) it's not dangerous.

Asbestos can be dangerous stuff, but there are lots of dangerous things in life. The issue is whether you have any exposure.

Before you start worrying about asbestos, where do you think it would be in your house that would pose a problem? In that vintage house, the only likely places are in vinyl tile or flooring, or if you have popcorn ceilings. Even if you have these products, there is no guarantee without testing that any asbestos is present, and, even if it is, as long as you aren't having it torn out, it's not an issue.

You should be worried about how you are going to afford to move out of the house you grew up in, not asbestos.