According to NASA a pulsar is a type of neutron star that rotates around an axis and emits a "pulse" of particles from the magnetic poles. This "pulse" consists of particles traveling close to the speed of light; as the star rotates the pulse also rotates, resulting in the emission of a bright beam of light, similar to a lighthouse emitting light perpendicular to the axis of rotation. Pulsars have been found in most wavelengths of the electromagnetic spectrum.
When answering the question of how pulsars can affect Earth we need to address two main variables: the distance from Earth, and the type of electromagnetic wave emitted. The greater the distance the less energy is transferred when the electromagnetic wave strikes an object. The more energy an electromagnetic wave possesses the greater it's capacity to affect a change in the medium it strikes. From our current body of knowledge it is not likely that a pulsar will affect Earth in a detectable way, primarily due to the distance from Earth.
To address the "how" part of the above question is then to begin a journey into speculation since the answer is that it is unlikely we will be affected. Since pulsars emit a pulse of electromagnetic radiation in the form of energy it is possible that this energy would affect Earth. The range of the electromagnetic spectrum is radio waves, microwaves, infrared, visible light, ultraviolet, x-ray, and gamma.
Everything below ultraviolet waves is pretty minimal risk and would require high concentration of waves and long exposure times to create a negative effect, so the result is that they primarily act as a beacon for detection. It is unlikely that if a pulsar was close enough to be concerned with that that the emission of these waves would be a cause for concern as our sun emits them all ready.
The waves above ultraviolet cause progressive damage the further you go. Beginning with ultraviolet radiation effects could be increased cancer, temperatures, basically everything we worry about with the sun. X-ray and Gamma rays would bombard the Earth's surface much more drastically causing death from radiation poisoning, broad scale mutations, and a high likely hood that life would not exist within those constant conditions of being bombarded by radiation.
The other possible affect that pulsars may have is on society and technology. Pulsars open the gateway of possibility to confirming our theories in physics like general relativity and gravitational waves. By having these beacons in space it offers us the opportunity to have map points while plotting stars, and ultimately technology could be influence as the theories in physics become laws. As technology increases so could society, thus providing a more esoteric sense of how a pulsar can affect Earth.