Well, like I said, it helps that the papers are forever blaming him for stuff that other people are responsible for, like with Spider-Man.
I guess this would work - if this newspaper were the only source of news in the world. Otherwise some paper, television reporter, or blogger is going to notice Green Hornet is doing good deeds and report that.
When the character was created in the '30s, newspapers
were the primary source of news in the world. They had a lot of influence over public opinion. In the '60s show, they modified it so that the
Daily Sentinel also operated a TV news station (with Gary Owens reporting from a studio right next to Britt Reid's office).
If the movie kept Reid as merely a newspaper publisher, then that represents a failure to modernize the concept plausibly, given that newspapers have nowhere near the relevance and impact today that they used to have. Maybe they should've made him the owner of a cable news network instead.