The problem with asbestos is that, due to its structure, it keeps breaking in the longitudinal direction, making thinner and thinner until it is the size of chromosomes. Physical interference with DNA is how it causes cancer.

My understanding is that basalt fibers seem to be glassy, not crystalline, so the breaking does not happen.

The source material is crystalline, but it becomes somewhat glassy during the fiber manufacture, when it is cooled too fast to crystallize completely.

Asbestos is made from some silicates (pyroxenes or amphiboles) which contain long covalent chains of silicon and oxygen atoms, which are more likely to separate than to break transversally.

Basalt also contains pyroxenes and amphiboles, but they are mixed with other kinds of silicates and they also have a different chemical composition than those of asbestos, so as far as it is known for now the probability of breaking into very thin fibers is very low for basalt fibers.

It is plausible that basalt fibers should be safer, because unlike with asbestos, which is made from rather rare minerals, basalt covers a large fraction of the surface of the Earth, so if basalt were dangerous erosion should have made harmful basalt fragments abundant in the environment.

My bartender told me the other night that as long as I have 2 cups of black coffee and some red meat before drinking, it will coat my liver and prevent cirrhosis. I'm wondering if I should have him check out the mole on my arm, just to put my mind at ease. He would probably volunteer to be an experimental subject to prove this glass theory.