The allusion is to Lewis Carol's 'Through the Looking Glass' in which a character of that name states to Alice:
"in this place it takes all the running you can do, to keep in the same place."
Cheetahs are almost certainly faster than their ancestors, and gazelles similarly improved relative to earlier versions. Despite this, today's cheetahs probably don't dine on fresh gazelle significantly more than previous generations. Both sides are 'fitter', but that hasn't changed the general dynamic. If either side were to gain too significant an advantage (e.g. cheetahs either catching everything in sight or starving), evolutionary pressure would act to bring things back.
We are seeing a number of seemingly Red Queen style arms races on today's Net - that between spammers and email clients, that between phishers (and other identity thieves) and unsuspecting email clickers, that between WAM products from competing vendors, etc.
For identity theft in particular, as defenses improve against existing attacks, the phishers/pharmers develop more elegant variations. Either side may enjoy a temporary advantage, but only until the other side catches up. If this race truly is a Red Queen phenomena, neither side will ever win out right and the general severity of the identity theft problem will remain relatively constant over time. Lets hope not.
p.s. One currently respected theory (as championed by Matt Ridley in his "Red Queen" book) for explaining the evolution (and relative ubiquity) of sex is that the genetic mixing that sexual reproduction enables (relative to asexual reproduction) gives hosts an advantage over the parasites and germs that attack them (the popularity of internet porn notwithstanding, sex does need an explanation because its costs relative to asexual reproduction are clear). By giving the genes a good churning with every generation, the theory posits that sex allows hosts to present a moving target to the pathogens that would prey on it. It's not clear to me what conclusions might be drawn from such a theory for the usefulness of sex for the battle between WAM products. Maybe being a product manager would be a better job than I've always believed.