Woodward and Bernstein were credible. That's the whole deal with anonymous sources: since the source isn't willing to stake their own credibility, you have to go on the credibility of the person doing the reporting instead. If a credible reporter/institution cites anonymous sources, I'm inclined to give a **** what they have to say. And they build that credibility by consistently getting the story right, and just as importantly by not getting stories wrong. If you want people to trust your claim that you have anonymous sources, you'd better have an established track record of vetting the hell out of them and being right.
If a zero-credibility hack cites anonymous sources, I'm inclined to bet they either made the whole thing up or are misrepresenting whatever sources they do have, who they probably didn't vet much or at all anyway. We just had a significant news story about a partisan source trying to deliberately feed false information to the Washington Post under the guise of being an anonymous source. WaPo vetted the source, caught them, and called them out on it. This is par for the course now. I would never argue that you should uniformly disbelieve anything sourced by anonymous sources. I will observe that you should trust claims attributed to anonymous sources exactly as much as you trust the entity reporting those claims. So if they're coming from ESPN, a place with a long-established track record of vetting nothing and running with agendas, **** em.