Freshly killed Voldemort (bodiless)

Whenever Voldy dies, his Hurcruxes keep him alive in a bodiless form.

I was ripped from my body, I was less than spirit, less than the meanest ghost . . . but still, I was alive. What I was, even I do not know . . . I, who have gone further than anybody along the path that leads to immortality.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

In this state a Baslisk would appear to have no effect because:

1. Voldy is already "less then a ghost" (who only get petrified when starring at Baslisks.)
2. Voldy possesses no body to get killed/petrified with.
3. By nature of being bodiless, Voldy has no eyes with which to unhealthy stare at Baslisks.

After asking magical mumbo jumbo

Voldy later used some weird, presumably dark, magic do give himself the famous form of a "hairless and scaly-looking", "dark, raw, reddish black" "crouched human child" with "thin and feeble" "arms and legs" and a "flat and snakelike" face, that we know and love throughout most of Goblet of Fire.

“Wormtail’s body, of course, was ill adapted for possession, as all assumed him dead, and would attract far too much attention if noticed. However, he was the able-bodied servant I needed, and, poor wizard though he is, Wormtail was able to follow the instructions I gave him, which would return me to a rudimentary, weak body of my own, a body I would be able to inhabit while awaiting the essential ingredients for true rebirth . . . a spell or two of my own invention . . . a little help from my dear Nagini,” Voldemort’s red eyes fell upon the continually circling snake, “a potion concocted from unicorn blood, and the snake venom Nagini provided . . . I was soon returned to an almost human form, and strong enough to travel.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

In this state, which possess a body and eyes, we can assume that looking at a Basilisk would either:

1. Kill loose this beautiful new body and downgrade Voldy back to being "less then spirit".
2. Petrify Voldy. (Similar to the effect on other mostly dead figures, e.g. ghosts)

The answer would depend on whether this new body is considered alive or not. I would go with the former, but considering the never seen before magic used to create this "body", I really have no idea.

Freshly killed Voldemort (bodiless)

Whenever Voldy dies, his Hurcruxes keep him alive in a bodiless form.

I was ripped from my body, I was less than spirit, less than the meanest ghost . . . but still, I was alive. What I was, even I do not know . . . I, who have gone further than anybody along the path that leads to immortality.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

In this state a Baslisk would appear to have no effect because:

1. Voldy is already "less then a ghost" (who only get petrified when starring at Baslisks.)
2. Voldy possesses no body to get killed/petrified with.
3. By nature of being bodiless, Voldy has no eyes with which to unhealthy stare at Baslisks.

After applying magical mumbo jumbo

Voldy later used some weird, presumably dark, magic do give himself the famous form of a "hairless and scaly-looking", "dark, raw, reddish black" "crouched human child" with "thin and feeble" "arms and legs" and a "flat and snakelike" face, that we know and love throughout most of Goblet of Fire.

“Wormtail’s body, of course, was ill adapted for possession, as all assumed him dead, and would attract far too much attention if noticed. However, he was the able-bodied servant I needed, and, poor wizard though he is, Wormtail was able to follow the instructions I gave him, which would return me to a rudimentary, weak body of my own, a body I would be able to inhabit while awaiting the essential ingredients for true rebirth . . . a spell or two of my own invention . . . a little help from my dear Nagini,” Voldemort’s red eyes fell upon the continually circling snake, “a potion concocted from unicorn blood, and the snake venom Nagini provided . . . I was soon returned to an almost human form, and strong enough to travel.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

In this state, which possess a body and eyes, we can assume that looking at a Basilisk would either:

1. Kill loose this beautiful new body and downgrade Voldy back to being "less then spirit".
2. Petrify Voldy. (Similar to the effect on other mostly dead figures, e.g. ghosts)

The answer would depend on whether this new body is considered alive or not. I would go with the former, but considering the never seen before magic used to create this "body", I really have no idea.

He would presumably return to his completely bodiless state.

After Voldemort died, his Hurcruxes kept him alive in a form, which by nature of being bodiless, had no eyes.

I was ripped from my body, I was less than spirit, less than the meanest ghost . . . but still, I was alive. What I was, even I do not know . . . I, who have gone further than anybody along the path that leads to immortality.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

He later used some magical mumbo jumbo to give himself the "hairless and scaly-looking", "dark, raw, reddish black" "crouched human child" form with "thin and feeble" "arms and legs" and a "flat and snakelike" face which was "a hundred times worse" than being "ugly, slimy, and blind", that he famously had throuhout most of Goblet of Fire.

“Wormtail’s body, of course, was ill adapted for possession, as all assumed him dead, and would attract far too much attention if noticed. However, he was the able-bodied servant I needed, and, poor wizard though he is, Wormtail was able to follow the instructions I gave him, which would return me to a rudimentary, weak body of my own, a body I would be able to inhabit while awaiting the essential ingredients for true rebirth . . . a spell or two of my own invention . . . a little help from my dear Nagini,” Voldemort’s red eyes fell upon the continually circling snake, “a potion concocted from unicorn blood, and the snake venom Nagini provided . . . I was soon returned to an almost human form, and strong enough to travel.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

Lacking any evidence to the contrary, we can assume that he would loose this beautiful new body and go back to being less then a spirit.

Freshly killed Voldemort (bodiless)

Whenever Voldy dies, his Hurcruxes keep him alive in a bodiless form.

I was ripped from my body, I was less than spirit, less than the meanest ghost . . . but still, I was alive. What I was, even I do not know . . . I, who have gone further than anybody along the path that leads to immortality.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

In this state a Baslisk would appear to have no effect because:

1. Voldy is already "less then a ghost" (who only get petrified when starring at Baslisks.)
2. Voldy possesses no body to get killed/petrified with.
3. By nature of being bodiless, Voldy has no eyes with which to unhealthy stare at Baslisks.

After asking magical mumbo jumbo

Voldy later used some weird, presumably dark, magic do give himself the famous form of a "hairless and scaly-looking", "dark, raw, reddish black" "crouched human child" with "thin and feeble" "arms and legs" and a "flat and snakelike" face, that we know and love throughout most of Goblet of Fire.

“Wormtail’s body, of course, was ill adapted for possession, as all assumed him dead, and would attract far too much attention if noticed. However, he was the able-bodied servant I needed, and, poor wizard though he is, Wormtail was able to follow the instructions I gave him, which would return me to a rudimentary, weak body of my own, a body I would be able to inhabit while awaiting the essential ingredients for true rebirth . . . a spell or two of my own invention . . . a little help from my dear Nagini,” Voldemort’s red eyes fell upon the continually circling snake, “a potion concocted from unicorn blood, and the snake venom Nagini provided . . . I was soon returned to an almost human form, and strong enough to travel.
_{(Harry Potter and the Goblet of Fire - Chapter 33)}

In this state, which possess a body and eyes, we can assume that looking at a Basilisk would either:

1. Kill loose this beautiful new body and downgrade Voldy back to being "less then spirit".
2. Petrify Voldy. (Similar to the effect on other mostly dead figures, e.g. ghosts)

The answer would depend on whether this new body is considered alive or not. I would go with the former, but considering the never seen before magic used to create this "body", I really have no idea.