First, it's really only an issue during the first 4 seasons of the new series. In the classic series, and in the post-Cracks era, nearly everything that takes place in the present is contained and easily covered up, or the events are somehow undone. There are a few exceptions, like the notorious "Nessie in the Thames", but those are explained in "Remembrance of the Daleks", when the Doctor tells Ace that the human capacity for self-deception prevents them from noticing things that don't fit in. So ultimately, people in the Whoniverse have no more reason to believe in alien invasions than people in the real world.
During the Russell T Davies era, things were different. He wanted to explore how events like a Dalek invasion would affect ordinary people, so he deliberately wrote stories that couldn't be contained or covered up. UNIT, Torchwood, etc. Clearly tried to keep them under wraps at first, but they soon had to give up. So, everyone knew about what was going on.
But Donna is a special case. It was a running joke on the show that she missed out on all of the big events that all of her friends and family kept talking about.
And Amy is also a special case. Her lack of knowledge of the Daleks was a major plot point—it was the first hint that the cracks in time were erasing events from history. And the fact that they were erased from history explains why she (and Rory, and Craig and presumably any future companions) don't know about them.
(If you consider the novels and other media canonical, the Virgin novels era was similar to the RTD era. But the Doctor's new companions—all from the future—knew about his more spectacular exploits as past history.)