Okay, my second cousin is working on a story that takes place on a planet named Odroia. It is an Earthlike planet with a supercontinent similar to Pangea, that has scattered rivers and lakes, with its global ocean having scattered islands, and it rotates westward like Venus. It also has a circumference of about 4040 miles, or 6472 km. All of that is irrelevant, as I'd like to talk the suns of Odroia. There's a nearby red sun and an distant blue sun. I've predicted that the skies would be a purple or magenta color, assuming an Earthlike atmosphere, due to a mixture of the light coming from both suns. Does my prediction shine, or is it all hot air?
Maybe... but very questionable.
As JBH already noted, our sky's color is mainly an effect of Raleigh scattering, which makes it blue despite our sun being yellow.
However, it isn't all blue; you do get some scattering of other wavelengths. If your red sun is very red, i.e. produces much more red light proportionally compared to blue light than our own sun, then you might get a sky that is somewhat more toward magenta. However, it will also probably be much darker than our sky.
I would suggest finding a paint swatch that is a fairly good match for our sky and looking at it in a) terrestrial sunlight (an analog for, well, our actual sky) and b) candlelight (an analog for a very low-temperature black body, which is what your red sun would be). In fact, just looking at the paint swatch in a variety of black-body lighting conditions will give you an idea for what colors of sky are most likely to occur.
If your small blue star is far enough away, it won't contribute enough light to have a significant effect. And you'll want it to be far away if a magenta sky is your goal.
That said, there's another possibility. Human eyes are really good at compensating for ambient lighting, which would tend to make the sky appear blue to us regardless of the actual balance of wavelengths being scattered. This is why colors look more or less the same under extreme variations in lighting conditions. However, if the vision of your inhabitants works differently, the sky might appear different to them regardless of its "absolute" color.
Logically it seems it would, but in reality, no
Rayleigh scattering, which is the mechanism that produces a blue sky during the day, favors the blue spectrum, principally due to the components of the atmosphere. Most of our atmospheric components scatter in the blue spectrum, despite our sun being yellow. Which is why we don't have a yellow sky.
I once found an article that demonstrated that all atomic gases scatter in the blue spectrum, which is why no atmosphere will appear anything other than a shade of blue. However, I can't find it again.
An excellent description of the base color issue can be found here: Terrestrial Exoplanet Skies – I've Built a Visual Sky Chart. Is it Correct?
Insofar as I'm aware, the only way to get a magenta sky is to saturate the atmosphere with something like an algae or pollen that reflects in the purple-red region. You might want to look through these site search results for questions about purple skies to see the various answers to the challenge.
I didn't vote this as a duplicate of any of those questions because a quick scan suggests none were specifically asking about a binary star system with red/blue suns. If someone finds that to not be the case and this question gets closed as a duplicate, let me know and I'll delete this answer.