NASA has used thermionics and thermoelectrics, but I've not heard of a major application of beta voltaics. Someone might be confusing technologies.
OK - there's a bit on alpha and beta voltaics at NASA Glenn.
Power & In-Space Propulsion
http://www.grc.nasa.gov/WWW/5000/pep/photo-space/
The problem of beta-voltaics is the spectrum of beta particles, and the gammas, and the decrease of activity with time. The activity is proportional to the number of atoms of radionuclide present and the decay constant. The greater the decay constant, the greater the activity for an amount of radionuclide, but that means it decays more rapidly.
Basically in beta-voltaics, there is a collector collecting the beta particles, so the source is at a + charge, which causes a potential difference across which a load is placed. There is a disadvantage with the beta spectrum such that some betas don't make it out of the source material, so the dimensions have to be properly sized based on the peak of the beta energy spectrum.