Habitable planets for Man, Stephen H. Dole, 1965
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
discusses the composition of an atmosphere habitable for humans. That should give some idea of changes which you can make to atmospheric composition.
Perhaps you can can make the atmosphere require breathing gear when humans venture outside to filter out a lot of red dust in the atmosphere. That red dust might make the sky seem red.
One thing you might do is make the world a giant moon orbiting a brown dwarf instead of a star, and the brown dwarf orbit a class M star emitting mostly orange light. The brown dwarf would emit mostly infrared radiation, and its visible radiation should be mostly reddish and orangish.
And possibly the heat on the surface of the world could come partially from the star's radiation, partially from the Brown Dwarf's radiation, and partially interal heat of hte moon resulting from tidal heating.
If much of the surface heat of the world comes from tidal heating, it may have many more active volcanoes than Earth. Perhaps the surface is only warm enough for life at the proper distance from an erupting volcano. The erupting volcanos will spread clouds of ash and dust for many miles around them, especially on the side the winds blow toward. And the red hot lava coming from the volcanos emits light reflected back down from the ash clouds to the ground and back up to the clouds, bouncing up and down and getting fainter with distance but still quite noticeable at the distances of all the human settlments.
And perhaps in the ares beyond the volcanic clouds where the sun shines brightly during the daytime, the ground is all red for some reason. Maybe it contains reddish iron oxides - rust. Maybe some process has covered the surface with red corundum (ruby) which would be highly reflective and reflect light upwards. And maybe there is a layer of dust or something in the atmosphere which reflects the red light reflected from the ground back down to the ground making the sky seem red.
The Tunguska event in 1904 produced a unusal effect.
Over the next few days, night skies in Asia and Europe were aglow.[13] There are contemporaneous reports of brightly lit photographs being successfully taken at midnight (without the aid of flashbulbs) in Sweden and Scotland.[11] It has been theorized that this sustained glowing effect was due to light passing through high-altitude ice particles that had formed at extremely low temperatures as a result of the explosion – a phenomenon that decades later was reproduced by Space Shuttles.[14][15] In the United States, a Smithsonian Astrophysical Observatory program at the Mount Wilson Observatory in California observed a months-long decrease in atmospheric transparency consistent with an increase in suspended dust particles.[16]
https://en.wikipedia.org/wiki/Tunguska_event
If it is too hot for humans to come out in the day, they might find the skies glowing at night from permanet ice crystals in the sky. And perhaps some process could make that glowing sky red.
If humans can only come out at night on the world in your story, they might see something like the zodiacal light seen from Earth, but many times brighter.
The zodiacal light (also called false dawn24 when seen before sunrise) is a faint glow of diffuse sunlight scattered by interplanetary dust. Brighter around the Sun, it appears in a particularly dark night sky to extend from the Sun's direction in a roughly triangular shape along the zodiac, and appears with less intensity and visibility along the whole ecliptic as the zodiacal band.[6]
The interplanetary dust in the Solar System forms a thick, pancake-shaped cloud called the zodiacal cloud which straddles the ecliptic plane. The particle sizes range from 10 to 300 micrometres, implying masses from one nanogram to tens of micrograms.[8][9]
https://en.wikipedia.org/wiki/Zodiacal_light
So the dust would have to be many times thicker to make a many times brighter zodiacal light. And some process would have to make the reflected light red.
The Star Trek song "Beyond Antares" says: "The sky is green and glowing, where my heart is."
Glowing sky reminds me of the aurora.
An aurora[a] (pl. aurorae or auroras),[b] also commonly known as the northern lights (aurora borealis) or southern lights (aurora australis),[c] is a natural light display in Earth's sky, predominantly seen in high-latitude regions (around the Arctic and Antarctic). Auroras display dynamic patterns of brilliant lights that appear as curtains, rays, spirals, or dynamic flickers covering the entire sky.2
Auroras are the result of disturbances in the magnetosphere caused by the solar wind. Major disturbances result from enhancements in the speed of the solar wind from coronal holes and coronal mass ejections. These disturbances alter the trajectories of charged particles in the magnetospheric plasma. These particles, mainly electrons and protons, precipitate into the upper atmosphere (thermosphere/exosphere). The resulting ionization and excitation of atmospheric constituents emit light of varying colour and complexity. The form of the aurora, occurring within bands around both polar regions, is also dependent on the amount of acceleration imparted to the precipitating particles.
Red: At its highest altitudes, excited atomic oxygen emits at 630 nm (red); low concentration of atoms and lower sensitivity of eyes at this wavelength make this color visible only under more intense solar activity.
Green: At lower altitudes, the more frequent collisions suppress the 630 nm (red) mode: rather the 557.7 nm emission (green) dominates. A fairly high concentration of atomic oxygen and higher eye sensitivity in green make green auroras the most common.
The Carrington Event was the most intense geomagnetic storm in recorded history, peaking from 1–2 September 1859 during solar cycle 10. It created strong auroral displays that were reported globally1 and caused sparking and even fires in multiple telegraph stations. The geomagnetic storm was most likely the result of a coronal mass ejection (CME) from the Sun colliding with Earth's magnetosphere.2
Auroras were seen around the world in both northern and southern hemispheres. The aurora over the Rocky Mountains in the United States was so bright that the glow woke gold miners, who began preparing breakfast because they thought it was morning.[8] People in the northeastern United States could read a newspaper by the aurora's light.[14] The aurora was visible from the poles to low latitude areas such as south-central Mexico,[15][16] Cuba, Hawaii, Queensland,[17] southern Japan and China,[18] and even at lower latitudes very close to the equator, such as in Colombia.[19]
https://en.wikipedia.org/wiki/Carrington_Event
So you would need the star to produce a constant stellar wind much stronger than the Carrington Event to provide constant world wide auroras.
And you might need a layer of dust in the atmosphere below the layers which produce green auroras, so it would block the green light from the ground and let only the red light through.
And I can think of a few other ways to make the sky red.
To be continued.
