[clueless (OP)]

Hello. There is a binary star consisting of two stars, red dwarf and blue dwarf. The red dwarf has a mass twice smaller than the mass of our Sun, while the blue dwarf is smaller than the red dwarf. There is a planet orbiting around the binary star, at a distance, say, close to the distance between Earth and the Sun, which planet is similar to Earth, with similar atmosphere. Life, apparently, in my story, evolved there, where live aliens similar to humans. What color is the sky during day? (Additionally, what color is the sky at the sunrise and the sunset?). Thanks a lot.

[Janus]

Blue.  The color of the sky is due to how it scatters light. 
A "red" dwarf doesn't just emit red light but in the the whole visible spectrum. It's spectrum is just more centered towards the red. It would appear to be light orange to our eyes.
A blue dwarf, also doesn't emit just Blue light, but has its spectrum shifted to the blue. It would appear to be very light blue. (though if you were to look directly at either star, they would overload your retina and they would appear white in the sky.)
The mixed light from each star would result in a more even spectrum ( the red star making up for the Blue and vice-versa).
This evens out the spectrum reaching the planet.
The blue part of the spectrum would scatter out giving you a blue sky during the day.
At sunset or sunrise, things could depend on the visual separation between the two stars, and which one sets first.
If the red star sets later, you could see a redder sunset, and if the Blue star sets later a paler sunset. 
But even here, the atmosphere plays a huge role.  More dust/smoke etc. in either situation will result in a more red sunset/sunrise.

[charles1948]

Hi, clue.  You raise an interesting question.  Firstly, I think you may have stumbled slightly by invoking "Blue Dwarfs".
As far as I'm aware, this class of stellar object hasn't yet been detected.  And would cause profound problems in theoretical astrophysics if it were.

I think what you mean is, suppose an inhabited planet, orbited a binary star system.   And the system comprised two stars of different colours.

Such as a Red Dwarf star, together with a white-yellow star like our Sol.  How would that affect the colour of the planet's sky?

I think, not much.  A more important factor would be the composition of the planet's atmosphere.

To take an example.  Before the planet Mars was well investigated by our space-probes, we knew, from Earth-based telescopic observations, that the Martian atmosphere was very thin.  Something similar to the Earth's atmosphere at a height of  50 or 60,000 feet.  And at that height, the Earth's atmosphere presents a dark blue, almost purple appearance.

So the general conception was that the Martian sky, as viewed from an observer on the surface of Mars, would be dark.

However this turned not to be true.  The Viking landers in 1976, showed that the Martian sky is surprisingly bright.
Not at all a dark purple colour.  Rather, a kind of mild brown sepia colour, like you see in 19th century photographs.

Why is this?  Because the Martian atmosphere contains myriads of dust particles, whipped up from the surface by unexpectedly strong winds.  The winds fling brownish dust up into the atmosphere, where it remains suspended for a long time.  It doesn't descend quickly, owing to the relative weak Martian gravity, less than a third of Earth gravity..

And while the dust is in the Martian atmosphere, it scatters light from the sun, diffuses the sunlight through the atmosphere, and so produces an very unexpectedly bright sky on Mars.

This Martian example leads me to conclude,  that the brightness/colour characteristics of the sky of an extra-solar planet, orbiting Sirius or Betelgeuse, or Vega, is more likely to be determined by the planet's atmosphere, than by its star.

[Halc]

There is a binary star consisting of two stars, red dwarf and blue dwarf. The red dwarf has a mass twice smaller than the mass of our Sun, while the blue dwarf is smaller than the red dwarf. There is a planet orbiting around the binary star, at a distance, say, close to the distance between Earth and the Sun

You seem to describe a circumbinary planet or P-type, where the planet orbits the binary pair (think Tatooine) rather than S-type where the planet orbits only one of the two stars.
A circumbinary planet needs to orbit at a distance of around say thrice the separation of the binary pair, else the orbit will not be stable. For a planet orbiting at 1AU, that puts the stars closer together than the distance from the sun to Mercury, and yet sufficiently separated that the stars don't tear each other apart, as is seen in some binary systems.

The weather will get interesting as the stars are aligned on a regular basis if the planet is in a similar orbital plane, and less regular basis if not.

Janus is correct, the color of the sky is more a function of the composition of the atmosphere and not so much a function of the spectrum of the star. Makes you wonder what color our sky was 4 billion years ago, before Earth became a 'class M' (star trek designation) planet with an unnatural oxygen rich atmosphere.

[Bored chemist]

Firstly, I think you may have stumbled slightly by invoking "Blue Dwarfs".

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

Sorry, I will get my coat.

[evan_au]

Binary star: What color is the sky on a planet?

I think the answer is pretty black and white.

As Halc implies, the orbit of most planets formed inside a binary star system are unstable; they mostly:
- Get flung off into interstellar space (in which case the sky is black)
- Get flung into one of the stars (in which case the sky and the planet dissolve into a white-hot plasma)

There are some stable orbits, but they are a minority.

[William Hardy]

The most common color for the sky on Earth is a white (or rather grey) sky. Clouds cover around 70% of the Earth, and nature by which they are scattering shortwave radiation makes them appear white or grey.

[Bored chemist]

The most common color for the sky on Earth is a white (or rather grey) sky. Clouds cover around 70% of the Earth, and nature by which they are scattering shortwave radiation makes them appear white or grey.

Do you think that posting other people's work makes you look good?
https://newbedev.com/could-the-sky-on-a-planet-theoretically-be-any-color