As long as they approach the body from the exterior, they are quite harmless to humans, because their interaction with matter is so intense that they are usually absorbed in the dead cells of the outer skin. Alpha rays emitted within the body, on the other hand, are dangerous.
Nuclei emitting alpha rays are usually very heavy, like Uranium or Plutonium.
There is also the proverbial alpha ray causing computer crashes. The background for this is that an alpha particle that manages to creep into memory chips (usually from cosmic rays, but impurities in the materials used to build the computer could also be radioactive) can deposit enough energy in the condensators? in DRAMs? to flip a memory bit.