Yes, the general public do indeed know about at least the most visible of the alien-related events shown in the series.
From Series 3 episode 1, Smith and Jones:
MARTHA: What do you think happened?
DOCTOR: What do you think?
MARTHA: Extraterrestrial. It's got to be. I don't know, a few years ago that would have sounded man, but these days? That spaceship flying into Big Ben, Christmas, those Cybermen things. I had a cousin. Adeola. She worked at Canary Wharf. She never came home.
Note that at this point Martha wasn't a Companion yet and had had no direct involvement in any of these events; she was just an ordinary Londoner, and was aware of all this extraterrestrial activity.
From Series 3 episode 11, The Sound of Drums:
SAXON: Britain, Britain, Britain. What extraordinary times we've had. Just a few years ago, this world was so small. And then they came, out of the unknown, falling from the skies. [Clip from ALIENS OF LONDON.] You've seen it happen... Big Ben destroyed, a spaceship over London. [Clip from ARMY OF GHOSTS.] All those ghosts and metal men. [Clip from RUNAWAY BRIDE.] The Christmas star that came to kill. Time and time again the government told you nothing. Well not me. Not Harold Saxon. Because my purpose here today is to tell you this... citizens of Great Britain... I have been contacted. A message, for humanity, from beyond the stars.
This was in a message broadcast on television to the entire United Kingdom, and took place before "the year that never was", so all these events are clearly common knowledge.
From the 2007 Christmas special, Voyage of the Damned:
WILF: Where have you been living? London at Christmas? Not safe, is it?
WILF: Well, it's them, up above (points skywards). Look, Christmas before last we had that big bloody spaceship, everyone standing on a roof. (points at his small TV that is showing a clip). And then last year, that Christmas Star electrocuting all over the place, draining the Thames.
Again, at this point neither Wilf nor Donna was a Companion, and Donna hadn't even told Wilf about the Doctor (she only did so in series 4 episode 1, Partners in Crime). So again, all these events are common knowledge.
From Series 4 episode 1, Partners in Crime:
DONNA (excited babbling): Cos I thought, how do I find the Doctor? And then I just thought, look for trouble and then he'll turn up! So I looked everywhere, you name it: UFOs, sightings, crop circles, sea monsters. I looked, I found them all. Like that stuff about the bees disappearing, I thought, I bet he's connected. Cos the thing is, Doctor, I believe it all now. You opened my eyes. All those amazing things out there, I believe them all. Well, apart from that replica of the Titanic flying over Buckingham Palace on Christmas Day, I mean that's gotta be a hoax!
Again Donna wasn't quite a Companion yet at this stage, but she'd heard of the events of Voyage of the Damned - and disbelieved them, interestingly enough.
So it seems that in the Whoniverse, humanity started being aware of aliens in the year 2005, and after that extraterrestrial events came thick and fast.
When the Doctor and his Companions travel back in time, the alien-related events that take place back then are always somehow hushed up (e.g. in The Shakespeare Code or Vincent and the Doctor). The first highly visible and non-retconned alien activity was apparently the Slitheen spaceship crashing into Big Ben in Aliens of London. Like the Doctor said in that episode:
DOCTOR: So maybe this is it. First contact. The day mankind officially comes into contact with an alien race. I'm not interfering because you've got to handle this on your own. That's when the human race finally grows up. Just this morning you were all tiny and small and made of clay. Now you can expand.
Which means that Big Ben crash has been reshown over and over, annoying me every time because it just looks so goofy and ridiculous. In fact it was designed to be so even in-universe! There were no real aliens inside; it was just a publicity stunt by the Slitheen, who were in fact hiding within the British government, to get attention for their own purposes. But that's neither here nor there, I suppose.