First, we should have some criteria on what to call a diploma mill.
This article cites the Ministry of Education claiming:
Indeed, the number of universities that worked like diploma conveyor line was growing in Russia, comments the Ministry of Science and Education. The Ministry and Federal Education Supervision Service are regularly doing checkups to identify the so-called diploma mills, with hundreds of students per one instructor or a tremendous ratio of extramural to intramural students -- these are one of the first hallmarks. Since 2015, the state accreditation has been renounced for 34 universities and branches. This mainly concerns Humanities, including Law, Economics and Management.
Original text
"Действительно, в России росло количество вузов, деятельность которых напоминала конвейер по выдаче дипломов,— прокомментировали эту историю в Минобрнауки.— Министерство совместно с Рособрнадзором регулярно проводит проверки, которые позволяют увидеть и так называемые конторы по продаже дипломов, когда на одного педагогического работника приходится одновременно несколько сотен студентов или же количество заочников в сотни раз превышает студентов очного отделения,— это одни из первых признаков. С начала 2015 года в результате проверок было приостановлено действие государственной аккредитации в 34 вузах и филиалах. В основном это касается укрупненных групп специальностей "Гуманитарные науки", в которую входят образовательные программы в области юриспруденции, и "Экономика и управление"".
An small sample of universities that get their accreditation renounced in early 2016 is mentioned by Federal Education Supervision Service:
For poor quality and content of education, the state accreditation is void for the following universities:
1. Kabardino-Balkariya business institute (founded 1998)
2. International Slavic institute (founded 1994)
3. Moscow Legal studies institute (founded 2000)
4. Moscow Institute of Banking (founded 1996)
5. Vladimir business institute (founded 1997)
6. Institute for Advocacy, Notariate and International Relations (founded 2003)
Original text
За неустранение несоответствий содержания и качества подготовки обучающихся полностью лишены государственной аккредитации:
1. НОУ ВПО «Кабардино-Балкарский институт бизнеса»;
2. ОАНО ВО «Международный славянский институт»;
3. НОУ ВПО «Московский институт юриспруденции»;
4. НОУ ВПО «Московский институт банковского дела»;
5. АНО ВПО «Владимирский институт бизнеса»;
6. НОУ ВПО «Институт адвокатуры, нотариата и международных отношений».
I think we're safe to assume these ones definitely were diploma mills. If you dig a bit more cases (1, 2, 3, 4, 5, 6), you see the trend. Diploma mills in Russia are either Moscow-based Humanities universities with very general names (I think they can teach you everything if they are called International Slavic Institute) or some distant region (North Caucasus, for instance) universities with very obscure names (I particularly liked the Institute for Psychology and Acmeology, whatever they could mean).
Now, these universities weren't there in the Soviet times. An overwhelming majority of them popped up in the 1990s (I had given the foundation dates for the six universities I listed above).
Back to your questions, the universities that were highly respected in the Soviet times continue to be legit universities. I should remark that the quality of education, from my insider point of view, seems to deteriorate, in the sense that students try to slack as much as they can, including giving bribes to instructors. However, no prominent university has turned to a diploma mill as in 'never come to the classes, pay and have your diploma'. The ones that are diploma mills are the newly emerging, mayfly universities.
Further reading
If you want to learn more about diploma mills in Russia, consider reading about Dissernet. While the state-based Federal Education Supervision Service is doing a good job fighting diploma mills that give bachelor and master's diplomas, Dissernet is a non-profit volunteer organization that studies plagiarism in PhD thesis works and busts PhD diploma mills.