Yes, the DNA encoded information is classic (please do not consider my comments as any kind of an endorsment of the OP article). But, as I said, this is only a half of the information you need to get any sensible predictions of the results. The rest is defined by folding, and it is very much quantum (heuristics aside).
> Proof?
I never heard of any polymerase simulation which did not involve computational quantum chemistry methods. Cannot even think of any possible classic approximation (again, heuristics aside - they're just masking the quantum nature of the underlying physics).
EDIT: as for transcription, do not forget about the operons. Theirs molecular mechanics is also very quantum.