Spin waves in antiferromagnets are promising alternative for replacing semiconductor technologies in future computational devices. One of the unique features of antiferromagnets is the availability of two spin waves with different chiralities—right-handed and left-handed ones. This degree of free-dom, in addition to their phase and amplitude, can be used to create modern computational devices. The search for an effective method to separate and control the two modes in antiferromagnets is now attracting the increasing attention. Previous studies have demonstrated that an electric field can manipulate antiferromagnetic magnons of different chiralities. However, the influence of the Aharonov–Casher effect on the damping right-handed and left-handed spin waves has not been fully investigated. In this work, based on Landau–Lifshitz–Gilbert equations with Rayleigh-dissipation functional, we show that the applied electric field could effectively control the damping of the spin waves with different chiralities. Temporal attenuation of the right-handed and left-handed modes has a shift along the wave-vector axis, which is directly proportional to the magnitude of the electric field. The relation between the damping parameters in the Rayleigh-dissipation functional plays an important role in dissipation behaviour. Our results show that the magnon lifetime depends on the electric field and dissipation parameters and is different for distinct chiralities. We believe our findings will encourage further investigation into dissipation processes in antiferromagnets and their impact on magnonic devices.