i believe there were two different concepts. the first one — an enemy of the state, or enemy of the people (враг народа) — is not about ethnicity. it's about "criminals" who deed horrible things to Soviet state and Soviet people.
some of these people were indeed criminals (although someone could argue that their crimes are usually not so horrible, i.e. banal theft), some of them — a lot, I think — were innocent.
enemy of the state/people was a working concept in 1930-1950s.
the second concept is about ethnicity. i don't know is there a name for it though. all of the ethnicities in Soviet Russia were declared equal, USSR was extremely anti-racist and anti-nationalist.
at the same time there were some nuances. for example, it was very hard to go to the top university if you are a jew (or jewish? I don't want to cause offence here). no one will tell you that you are not qualified because you are jewish, but you won't be there anyway.
there were two reasons behind that. the first one: it was too dangerous to give a good education to someone who could leave USSR for Israel later. the second one is quite obvious, i think.
I don't know when it began but it was relatively difficult for jewish kids to get to the top faculties of top universities even in 1980s. the same could be said about top-secret research and some other areas.