Ironically, people used the exact same argument against the printing press and pretty much every single development in publishing. Remember when the gramophone destroyed the live performance? When TV killed theatre?
All of those developments lowered the price of production and/or distribution radically, and allowed the production of new cheap crap, but did not kill the incumbent as predicted.
The same is true today. Before the Internet, you would probably be subscribing to a newspaper, probably at quite high expense. If you limited yourself to free reading, you would only get crap - as true then as now.
If the traditional newspaper is largely irrelevant, there's a large and healthy selection of paid-for journalism out there, just get out your wallet. I'm a fan of The Economist, but that's far from the only selection.