How can I minimize the damage of my PhD advisor's bad recommendation when applying for industry jobs? I did not get along well with my former PhD advisor. I was able to defend and graduate; however, I have learned that my advisor was giving bad recommendation letters. No wonder why all of my postdoctoral applications were unsuccessful even though I have good academic standing. Since we had several intense arguments in the past, I can conclude with confidence that any reference from my advisor would be damaging.
I have decided to go to industry (non-research) not just because of this problem but also because of poor job prospects in academia. Now, I am wondering how I can minimize the damage. I do not know how references work in industry, but I can certainly exclude my advisor in any list throughout a recruitment process. My PhD advisor also has several connections in HR. This can also be damaging if a company background-checks through the university HR department.
You are worried to much.

My PhD advisor also has several connections in HR. This can also be
damaging if a company background-checks through the university HR
department.

First, HR is not allowed to say anything bad about you with no clear evidence (for fear of being sued). They will just give minimal information, and that's it.
Second, background check is often done by a third-party, who is only interested in ticking all the boxes that you declared. They are not interested in hearing some random stories.

I do not know how references work in industry, but I can certainly
exclude my advisor in any list throughout a recruitment process.

For non-research positions, nobody care about references.