If they are telling the truth about the Pfizer and Moderna vaccinations' mechanism of action, then it does not alter DNA. The vaccinations, according to them, are pieces of mRNA which when enter the host cell, cause your own cells molecular machinery to produce proteins. This all occurs in the cytoplasm, away from your cell's own DNA. It can still be dangerous, however. I'm not saying its not dangerous. But, theoretically, it should not affect the structure of your own DNA.