My hazy memory of recent history of newspapers&the internet is that as the dot com boom got going, many newspapers did try to embrace the internet and spent time and money thinking about how to do that. Then when the dot com bust hit there was an almost relieved "oh, good, you can't make money on the internets anyway" and such efforts were scaled back.
But, basically, newspapers could have done anything on the internet that anyone else did. I think there was some understanding of this early on, with Knight-Ridder and others setting up what were poised to be branded local portal sites (philly.com, boston.com) but which sort of stagnated creatively, both in terms of content and potential revenue streams. The point is that they had fairly high traffic web sites and known brands and they could have leveraged that for lots of things. Free classifieds, online dating, concert and other event ticket sales (either directly or commission cut), affiliations with online retailers like Amazon, local search. Whether or not any of these or better ideas could provide enough revenue to support a rough approximation of their news gathering operation I don't know, but you got the sense that many papers weren't even trying.
And we shouldn't leave aside the issue of content, both in print and online. It seems that newspapers are partially hamstrung by outmoded notions of what print journalism is supposed to be and, frankly, they should think a bit harder about giving the people what they want. Failure to change was in part due to the notion that they were important civic institutions giving the city what it needed instead of what people wanted, which worked as long as the monopoly held. That doesn't mean dumbing things down or trying to be "hip," it means making it more interesting. People like stories. Lots of stories in my big dumb city. Tell interesting stories! The Philadelphia Inquirer should be more like The Wire and less like wire copy.