How far from Betelgeuse is its habitable zone?

Nobody knows the limits of the Sun's habitable zone, or how broad or narrow it is.

Here is a link to a list of various estimates of the inner, or outer, or both, edges of the circumstellar habitable zone of the Sun:

https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates

Note that one well known estimate, that of Hart et al. in 1979, makes the habitable zone very narrow, while another well known estimate, that of Kasting et al in 1993, produces a conservative habitable zone that is several times as wide as Hart's and an optimistic habitable zone that is wider still.

And there are other estimates with the inner edge of the habitable zone closer to or farther from the Sun than Kasting's and with the outer edge of the habitable zone closer to or farther from the Sun than Kasting's.

So if you study all of the original papers where those limits of the habitable zone were proposed, you can decide which ones are most convincing to you, and then use the habitable zone for the Sun to compare with Betelgeuse to determine Betelgeuse's habitable zone.

Taking the safe route, I will assume that a planet would have to receive exactly as much radiation from Betelgeuse as Earth receives from the Sun in order to be habitable.

According to Wikipedia, Betelgeuse has a luminosity about 126,000 times that of the Sun.

https://en.wikipedia.org/wiki/Betelgeuse

Since the square root of 126,000 is 354.96, a planet orbiting about 354.96 AU from Betelgeuse should receive the same amount of radiation from Betelgeuse as Earth receives from the Sun and thus should be within the circumstellar habitable zone of Betelgeuse, no matter how broad or narrow that habitable zone is.

But Betelgeuse is a variable star. Its luminosity varies between about 90,000 to 150,000 times the luminosity of the Sun, so the distance from Betelgeuse where a planet would receive exactly as much radiation from Betelgeuse was Earth receives from the Sun would vary between 300 AU and 387.298 AU.

Possibly a planet could orbit Betelgeuse with a somewhat elliptical orbit so it was closest to Betelgeuse when Betelgeuse was least luminous and farthest from Betelgeuse when Betelgeuse was most luminous and so constantly receive the same amount of radiation as Earth receives from the Sun.

Betelgeuse is classified as a semiregular variable star, indicating that some periodicity is noticeable in the brightness changes, but amplitudes may vary, cycles may have different lengths, and there may be standstills or periods of irregularity. It is placed in subgroup SRc; these are pulsating red supergiants with amplitudes around one magnitude and periods from tens to hundreds of days.[8]

https://en.wikipedia.org/wiki/Betelgeuse#Variability

The semiregular nature of Betelgeuse's variability means that nobody could even begin to design an orbit around Betelgeuse that would enable a planet to constantly receive the same amount of radiation as Earth receives from the Sun.

So planets in the habitable zone of Betelgeuse would receive significantly varying amounts of radiation from Betelgeuse as the star varied. Whether that would make life impossible on those hypothetical planets is unknown.

I may add that there are still more problems with having a planet with conditions suitable for life orbiting Betelgeuse; the answer by antispinwards mentions some of them.

So the probability that there will be lifeforms on any planets orbiting Betelgeuse when Betelgeuse becomes a supernova and destroys all its planets seems to be very, very, very low.

I note that humans can not survive in the majority of the biosphere of Earth. Humans can't survive in most places where other Earth lifeforms can survive.

A discussion of the conditions necessary for humans or similar beings to survive can be found in Habitable Planets for Man Stephen H. Dole, 1964, 2007.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

Here is a link to a list of the nearest stars and brown dwarfs to the Sun, stars within a distances of 5 parsecs or 16.3 light years:

https://en.wikipedia.org/wiki/List_of_nearest_stars_and_brown_dwarfs

There could be worlds with some kind of life in some of those star systems.

And what would happens to those hypothetical worlds with life orbiting stars within 5 parsecs or 16.3 light years of the Sun if the Sun became a supernova? The Sun will never become a supernova, but if it did the planets of those nearby stars would probable receive so much radiation during the supernova event that their oceans and atmospheres would boil away and their surfaces turn to red hot lava, and all life on them would die.

And a supernova might be deadly to life on worlds at a much greater distance. I am not very familiar with the distances at which a supernova would wipe out all life on a planet.

Betelgeuse is expected to have a supernova explosion in less than 100,000 years. During that time many stars will get closer and closer to Betelgeuse and then start to get farther and farther from Betelgeuse, as those stars and Betelgeuse orbit around the center of the galaxy.

Here is a link to a list of stars which, according to the calculations of astronomers, have passed within 5 light years of Earth in the past three million years or will pass within 5 light years of Earth in the next three million years.

https://en.wikipedia.org/wiki/List_of_nearest_stars_and_brown_dwarfs

And it seems to me that any alien astronomers on a planet orbiting a star near Betelgeuse would be very displeased to learn that their star is getting closer and closer to Betelgeuse and is likely to be only a few light years from Betelgeuse when Betelgeuse supernovas.