What if the Terminator was wrong, and Skynet doesn’t plan to eliminate humans via a nuclear apocalypse, but through a global pandemic?
One way that Skynet can kill us all
Below is a photo of the electro-mechanical soap dispenser in the bathroom of my office building. Notice how it has dismantled itself, so soap is no longer dispensed. An entire floor of workers has stopped washing their hands, and it’s just a matter of time before someone gets sick.
Was this faucet designed by Cyberdyne?
As Jude Law learned the hard way (see photo below) in Contagion, it just takes one sick person to start a massive epidemic, killing off vast quantities of humans. If (when?) Skynet becomes self-aware, how do we know that the smoking gun will be in the shape of a mushroom cloud? Maybe it will be in the shape of a sneeze or a cough, a high fever and some agonal spasms.
Another way for Skynet to wipe us out
Thoughtbasket 
Very scary, Thoughtbasket!