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. Since they were talking about the implications for a precisely engineered ecosystem, Geordi would have/should have been aware of Martin's concerns. At least with respect to Star Trek
transporters of the 24th century, "Won't affect her DNA at all" means just that, and therefore that her risk of any illness from genetic damage whatsoever wouldn't be measurably increased, by just the few transports they expected that Hannah would need to undergo.