By "does no DNA damage", they mean "won't turn you into a salamander".
I disagree. I think that Martin of the Masterpiece Society would have been concerned with even a slight increase in the risk of Hannah getting, say, cancer.
Why? Why would you think that?
Besides, even if the transporter does do damage at the genetic level, the approaching stellar core fragment doesn't allow them time to debate the matter or calculate risks and probabilities. Never underestimate how many times the characters lie, exaggerate, or soft-pedal the truth.