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 109 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).