It seems most likely that it would Petrify the spirit-form Dark Lord.
The Dark Lord, when he’s ripped from his body, existed in a spirit form. He can’t be truly killed while his Horcruxes ensure he stays earthbound, so looking at a basilisk wouldn’t kill him as he couldn’t die. Having Horcruxes meant his soul couldn’t ‘move on’ while they were intact - he couldn’t die.
“Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged. But, of course, existence in such a form …”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
Since the Dark Lord after being ripped from his body is similar to a ghost (though not quite one), it seems most likely that he’d be Petrified similarly to how a ghost would be.
“It was Nearly Headless Nick, no longer pearly-white and transparent, but black and smoky, floating immobile and horizontal, six inches off the floor. His head was half off and his face wore an expression of shock identical to Justin’s.”
- Harry Potter and the Chamber of Secrets, Chapter 11 (The Duelling Club)
Though it’s in no way certain that the Dark Lord would be Petrified if he looked at a basilisk while in spirit form, it seems to be the most likely result from what’s known about what happens when other types of spirits look at a basilisk. He seems clearly able to see while in spirit form, since right after he’s ripped from his body, he sees the Potter house had turned to rubble and Harry was trapped. He was only just ripped from his body, so wouldn’t have had a chance to possess anything else yet.
And then he broke: he was nothing, nothing but pain and terror, and he must hide himself, not here in the rubble of the ruined house, where the child was trapped and screaming, but far away … far away …”
- Harry Potter and the Deathly Hallows, Chapter 17 (Bathilda’s Secret)
This shows that the Dark Lord was able to see in spirit form, so he would be capable of looking at a basilisk. Since he can’t die, and other spirits get Petrified, it seems most likely he’d get Petrified.