And how do they do that, precisely?

It will vary from place to place, community by community. One technique that worked well for the GOP here in Florida was one of their efforts to increase support among Latino voters. They went hardcore into those neighborhoods, years before the elections and outside of campaigns, and opened up community centers to just do constant outreach to those voters:

nationalreview.com - Gop hispanic community centers are a step in the right direction

I have a "Trump" filter on my news, but still I get enough stories (POTUS is an important news item, and is in a lot more stuff than one might expect) to see the disaster and chaos, and know that almost anyone would be a better choice.

I don't think the Democrats can get the change they want through "news" coverage. They need to communicate through non-news channels. And channels that are consumed by people who aren't already already Democrats. That's a tall order. I don't know how you would go about building that up, but that's where the political battles of the next few decades are going to be fought. The age of political pamphlets moved to the age of newspapers to the age of radio to the age of television - and now it's moved on to whatever it is the current generations are doing. So the Democratic party needs to get there.