I looked at the original paper, Blank and Goodman "DNA is a fractal antenna in electromagnetic fields" Int J Radiation Biology, 87:4, 409-415, DOI:10.3109/09553002.2011.538130.
To be honest I can't give concrete reasons to doubt it, but here are a few things that raise my suspicions:
- it is a review article that relies heavily on self-citation (15/50 citations are authored or co-authored by the two authors of this paper, including most of the primary experimental references)
- in my judgement "responding to many different frequencies" and "having structures on several different scales" (Table 1) is not particularly strong evidence to support the conjecture that DNA acts as a fractal antenna; I would be much more convinced by an analysis based on physics rather than analogy (this comment on the paper by Foster reviews the physics of fractal antennas and its application to DNA and concludes that "Loose and implausible conjectures about DNA as a fractal antenna do not substitute for careful discussion of these matters"; I don't find the authors' rebuttal particularly convincing)
- the authors cite epidemiological evidence that non-ionizing/low-frequency radiation causes cancer: according to the US NIH's National Cancer Institute, this evidence is weak. (However, one of the authors of this paper is a contrarian on this subject and believes that the mainstream view is wrong, for a variety of reasons.) (See Are low-intensity radio-waves carcinogenic? for more discussion on Biology.SE.)
- the paper uses very disparate lines of evidence (not necessarily a bad thing, but in this case it feels haphazard): in particular, the authors discuss both low-frequency and ionizing radiation (this is part of their "DNA is sensitive to many different frequencies" argument, but ionizing radiation operates in a very different way)
- it is highly speculative in places (e.g. "EMF is believed to have been an important driving force in evolution", p. 413, no reference; the authors go on to attribute the faster evolution of eukaryotes to the fact that their DNA structures are more fractal)
- at least one of the references cited (de Pomerai et al. Nature 2000) was retracted in 2006 (5 years before the current paper was published)
What I can say in favor of the authors is that exploring the mechanisms by which low-frequency electromagnetic fields/radiation (ELF) affect cellular biology is indeed interesting; many of the epidemiological studies, while finding very weak effects, have also downplayed risk because so little is known about mechanisms of operation. If we could come up with a rigorous mechanistic understanding of ELF effects, that would be scientifically worthwhile.