What will happen if close stars like Sirius A or Procyon_A become red giants? Is it dangerous for the Earth and life on it?
If for Sirius A we still have 500-800 million years, but Procyon A might become a red giant in 10 to 100 million years, by astronomical scale it is very soon.
Or nothing will happen with the Earth, because it is still too far?
Nothing will happen. Neither star is massive enough to become a supernova and their velocities relative to the Earth mean that we will almost certainly be hundereds of light years away from them when they become red giants.
A 1 km/s velocity difference over 100 million years leads to a distance difference of about 300 light years.
Both Procyon and Sirius are in binary systems with white dwarfs, but they are not close binary systems. It is possible, but unlikely, that their white dwarf companions could accrete sufficient material to cause a Type Ia supernova - a possibility I address in Will Sirius B start accreting from A and become a supernova type Ia? but even then, as I said above, these stars will be nowhere near the Sun when that happens.
Life on Earth will not be adversely affected when nearby stars become red giants.
Certainly, things will get very messy inside those systems: red giants throw out a lot of gas and dust. Red giants are a lot more luminous than the Sun because they have such a huge surface area, but they don't emit large amounts of dangerous radiation, like X-rays or ultraviolet light. In fact, a red giant is cooler than the Sun, so it emits relatively little UV light.
According to Wikipedia, it's possible that a red giant system could even be habitable, so life in nearby systems should be quite safe.
Although traditionally it has been suggested the evolution of a star into a red giant will render its planetary system, if present, uninhabitable, some research suggests that, during the evolution of a 1 M☉ star along the red-giant branch, it could harbor a habitable zone for several billion years at 2 AU out to around 100 million years at 9 AU out, giving perhaps enough time for life to develop on a suitable world. After the red-giant stage, there would for such a star be a habitable zone between 7 and 22 AU for an additional billion years. Later studies have refined this scenario, showing how for a 1 M☉ star the habitable zone lasts from $10^8$ years for a planet with an orbit similar to that of Mars to $2.1×10^8$ yr for one that orbits at Saturn's distance to the Sun, the maximum time ($3.7×10^8$ yr) corresponding for planets orbiting at the distance of Jupiter. However, for planets orbiting a 0.5 M☉ star in equivalent orbits to those of Jupiter and Saturn they would be in the habitable zone for $5.8×10^9$ yr and $2.1×10^9$ yr respectively; for stars more massive than the Sun, the times are considerably shorter.
However, the Sirius and Procyon systems mentioned in the question are both binary systems, and both Sirius B and Procyon B are already white dwarfs. That makes the picture more complicated, and definitely more dangerous inside those systems, but neighboring stellar systems won't be harmed.
When Sirius A and Procyon A become red giants their white dwarf companions will get bombarded by a lot of material. If a white dwarf accretes enough hydrogen, it can lead to a runaway fusion reaction, in other words, a nova explosion.
Hydrogen fusion may occur in a stable manner on the surface of the white dwarf for a narrow range of accretion rates, giving rise to a super soft X-ray source, but for most binary system parameters, the hydrogen burning is unstable thermally and rapidly converts a large amount of the hydrogen into other, heavier chemical elements in a runaway reaction, liberating an enormous amount of energy. This blows the remaining gases away from the surface of the white dwarf surface and produces an extremely bright outburst of light.
A nova is bright, but it only emits a tiny amount of energy compared to a supernova, and poses no danger to neighboring stellar systems.
Actually, it's very unlikely that either Sirius B or Procyon B will have nova events: they are simply too far from their companions to accrete enough hydrogen for that to occur. (Thanks to Peter Erwin for that info).
