Harry Potter thinks the headmaster of Hogwarts is the greatest thing since sliced bread and, truth be told, he's not alone. Lots of witches and wizards trust Dumbledore to keep their kids safe.
Of course that doesn't mean Dumbledore always delivers. Sure, no one died at Hogwarts this year, but the several students were attacked by a basilisk and one was kidnapped, so we're pretty sure most educators wouldn't call that their most successful year.
And what's more, Dumbledore knows after the second attack that things are getting real:
PROFESSOR MCGONAGALL: What can this mean, Albus?
ALBUS DUMBLEDORE: It means that our students are in great danger.
PROFESSOR MCGONAGALL: What should I tell the staff?
ALBUS DUMBLEDORE: The truth. Tell them Hogwarts is no longer safe. It is as we feared, Minerva. The Chamber of Secrets has indeed been opened again.
Okay, so if Hogwarts is no longer safe, then you'll be closing the school down, right? Nope. Dumbledore keeps this ship running even though he knows last time the Chamber of Secrets was opened a student died.
What on earth is he doing? Does he know the monster is a basilisk? Does he know Voldemort is responsible? Does he have any idea where the entrance to the chamber is? After all, a couple of twelve year olds ultimately figure it out so you'd think "the greatest sorcerer in the world" who'd had all this info for the last fifty years would have a jump start on them.
But we digress.
In the end, Harry has faith in Dumbledore and that's what helps him in the Chamber of Secrets. His loyalty to Dumbledore calls Fawkes to him, which turns out to be a literal lifesaver. And even though we get the feeling that Dumbledore isn't telling us everything in that office scene, Harry doesn't seem to mind. He still sees Dumbledore as the wise mentor figure at this stage
We aren't really supposed to question Dumbledore because Harry doesn't question Dumbledore. At least not until, like, at least the fifth movie.