The camera’s insides would likely melt and not show anything.
Colin Creevey was using his camera when he encountered the basilisk. However, instead of recording any pictures of the basilisk, the camera was melted and damaged.
“Dumbledore didn’t answer. He prised open the back of the camera.
‘Good gracious!’ said Madam Pomfrey.
A jet of steam had hissed out of the camera. Harry, three beds away, caught the acrid smell of burnt plastic.
‘Melted,’ said Madam Pomfrey wonderingly, ‘all melted …”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)
This seems to indicate that it’s impossible to capture a basilisk on camera.
It’s unclear what would happen to the one controlling the camera.
Colin still got Petrified when looking at the basilisk through the camera. But this doesn’t necessarily mean someone using a remote camera would also be Petrified. Colin was looking through the lens, seeing the basilisk himself. If the remote camera melts as soon as the basilisk looks at it, then whoever’s watching will just see nothing.