Multiple star systems are common in the universe. Multiple star systems can take many forms, from a larger star with a smaller star orbiting it, to two larger stars orbiting each other, to nested stars where a star orbits a group of two stars orbiting each other, and so on. Multiple star systems can contain main sequence stars, giant stars, or degenerate stars. The possibilities of life in a multiple star system have been investigated in fiction, such as in Isaac Asimov's short story Nightfall, and the desert planet Tatooine's twin suns in Star Wars.
Life around a binary star seems like a rather extreme affair, so I decided on a thought experiment to see what it would be like to live in such a system. What would happen if we increased the size of Jupiter by 100, making it a small red dwarf star, about the size of Proxima Centauri. Increased by 100, the mass of Jupiter would be 10% of the mass of the sun, able to form a hydrogen fusing star. It would be about 2 or 3 times as large in radial dimensions, and 10 to 30 times larger in volume. What would life be like on our earth, in a solar system with two stars?
The gravity of this new star wouldn't be that great, not enough to have an obvious effect on the earth. This red dwarf has 10% of the mass of the sun, meaning that it has 10% of the gravity. Jupiter's orbit is around 5 Astronomical Units from the earth, meaning that the closest it approaches is 4 AU from the earth. Since, at its closest, it is four times as far as the sun, and since gravity decreases with the square of the distance, it would have a maximum of 1/160th of the gravitational pull of the sun. It would therefore have a small effect on our tides, but it wouldn't obviously change our orbit much. (But I will talk about some of the indirect effects below).
What about light and heat? Would the earth lack a night, and be scorched by two suns, never having a true night? The answer is that a red dwarf in Jupiter's orbit would not provide any measurable heat, but would provide visible light that would probably be somewhat like moonlight. Stellar output drops off very steeply with declining size, and their light becomes redder, with much of it in the infrared. Including infrared light, a red dwarf star might only be one part in 1000 as bright as our sun. Counting only visible light, it would be closer to one part in 10,000. But, just as with gravity, this red dwarf is further away from us than the sun is, meaning those numbers would go down by at least a factor of 10. We can look at a red dwarf star in Jupiter's orbit as being 0.01% as bright as the sun. This might seem like very little, but this is still brighter than a full moon, which is 1/400,000th as bright as the sun. Our red dwarf would then (when it was close), fill our night skies with a shadowy red light that would not provide any heat, and would be less bright than the ambient light of a city, but would still make the night visible enough to move and even read in. And much like the moon, the light of our red dwarf would be visible in the day (not too much of a surprise, since even Venus can be visible during the day). But unlike the light of the moon, the light from the red dwarf wouldn't be seen as coming from a disk, but rather from a point like light source. Jupiter isn't resolvable as anything but a point, and increasing its visual dimensions by 2 or 3 times probably wouldn't change that. It might have a stellar corona around it, which would increase its size until it was visible as an extended region, but even that would be barely visible.
In many ways, life on a planet in a system with a red dwarf would be pretty normal. However, the butterfly effect would certainly be in play in such a situation. Most scientists agree that Jupiter has a big role on the solar system, by attracting and absorbing comets and asteroids, as well as perturbing the orbit of the earth, leading in part to the Milankovich Cycles. Replacing Jupiter with something 100 times as large, even if it wouldn't automatically throw the solar system into a chaos of collisions and radiation, would still probably have changed the way life evolved on earth, including, most noticeably, changing several mass extinction events. Life would have probably evolved early on to see further into the infrared, taking advantage of the faintly illuminated night sky.
But on the whole, not only can we imagine life managing to evolve on an earth like planet with a red dwarf orbiting further out in its system, and if the system managed to settle down into relative stability, intelligent life there might adapt that not only adjusted to a two star system, but wasn't even particularly aware or concerned with living in a two star system.