It would probably be quite difficult to kill a petrified person, and doing so might not be sensible.
The word “petrify” is usually used to mean “turn to stone” (originally from the Greek word petra, meaning rock). Although I don’t think the condition is explicitly defined in the Harry Potter canon, we have no reason to think the definition is different from convention.
(My answer to Besides the Basilisk, what kind of magic can cause petrification? has some fairly flimsy canon about Gorgons, which link petrification and turning somebody to stone. That’s the best I can think of.)
It’s probably quite hard to kill a person who’s been petrified, if they have truly been turned to stone. There would need to be a link between the life of the original person and the status of the stone.
About the only thing I can think of would be to damage or destroy the stone, but if there was a way to repair the stone before reconstituting the person, then it would probably be for naught. If you were thorough, then you could grind it into dust and scatter the dust to the four corners of the earth, but that’s beyond a Basilisk’s capabilities.
I’m sure the Basilisk could smash a statue (perhaps by throwing it against a wall), but it might not be sensible to do so. Not unless it could cart the person off first, but that would detract from the intimidation aspect.
That would cause noise and attract unwanted attention. It might also make it easier to deduce that it was a Basilisk at work. The class of creatures which can petrify somebody is sufficiently large that a Basilisk isn’t the only candidate; but attacking the petrified person would narrow the pool. Part of the Basilisk’s advantage is the element of surprise; giving clues to its identity would be unwise.