Probably not, DNA degrades relatively quickly in uncontrolled conditions, so the unique markers of an individual human would probably be hard to grab from a dust particle suspended on a spider web. This works for detecting different species, not necessarily individuals.
Plus, there are infinitely easier ways of surveilling humans, like just watching where their GPS tracked phones go.
There are also easier ways of tracking humans via DNA - we're literally shedding genetic material with every step, on everything we touch. Skin flakes here, saliva there, etc.
So I guess the cyberpunk equivalent of TFA: disassembling a router, wireless phone charger, or any of the countless devices equipped with a tiny cooling fan, that increasingly live among us in our homes and offices, in order to sample DNA from the dust that accumulates on the fan blades.
(To add a little insult to injury: pull it off an air quality monitor. Those PM sensors have a neat, tiny, dust-moving fan.)
Plus, there are infinitely easier ways of surveilling humans, like just watching where their GPS tracked phones go.