Why do we know that the right-handed electrons exist? I am wondering how come Weinberg and Salam put the right-handed electrons and did not put right-handed neutrinos* into the theory? In other words, are there any experimental/theoretical indications that right-handed electrons exist?
*I know that the right-handed neutrinos were never observed and e.g. Goldhaber experiment excluded they exist with a high certainty.
 A: The moment you have a (Dirac; their charge prevents a Majorana one!) electron mass, you need right-chiral electrons, as you need both chiralities for such a term to exit. Electrons are massive, and so you need both right and left electrons in a realistic  theory.
At the time, neutrino masses had not been inferred, as they have been now, through neutrino oscillation experiments. A mere historical accident. For massless neutrinos,  W & S  did not need right chiral neutrinos (which are sterile, i.e. they do not interact weakly), so they succumbed to Occam's razor and did not bother introducing something nobody had seen at the time. 
This should be the full answer to your question, but I'd be remiss if I did not seize the teaching moment: Had they known about neutrino oscillations and masses, they would have incontrovertibly and trivially introduced the analog Higgs-coupling originating masses for them, formally identical to the one giving mass to the up-type quarks. In that sense, right-handed neutrinos are as part of the Standard Model as charged leptons are: they run on its ingenious gauge structure. So, the purported BSM status of them is a bit of a hollow and  irresponsible hype flash. The original ("cockeyed", Feynman called it) standard model was sufficiently outlandish to avoid needless conjectural extensions to even more new particles/states.
