Although the accusation that the Ministerium für Staatssicherheit (MfS) consisted of former security personnel of the Third Reich dates back to the very inception of the MfS in 1950, for the most part, very few of the MfS's personnel came from that particular cohort. Unlike the SS's RSHA, the MfS did not recruit its initial members from existing police personnel or from young professionals. Instead, the first generation of the MfS had a very bifurcated composition with regards to age. The MfS's leadership positions were largely composed of old-fighters from the Weimar-era KPD. Most of these men were veterans of the Spanish Civil War and had survived the various internal purges while in exile. The mass of recruits for the new agency were largely drawn from the GDR's youth population. By 1952, about half of the MfS personnel were under their thirties. While this youth cohort was young enough to experience the Third Reich, they did not fit the typical social profile for RSHA membership.

The recruitment among the GDR youth was part of a deliberate strategy by the MfS to avoid inadvertently recruiting moles into this new agency. Part of the SED's rationale for forming such a large security organization was that there was a fear that former Nazis would work to undermine the GDR from within. In this scenario, former members of the NSDAP worked in tandem with Western intelligence agencies. The concern over former Nazis acting as a fifth column was one which dated back to the period of Soviet occupation. When the Soviet Military Occupation Government (SVAG) created K-5, a department of the Eastern zone's German police charged with monitoring internal politics and a precursor of the MfS, SVAG explicitly ruled out anyone with prior membership in either the RSHA or SD from membership in K-5. The MfS continued many of SVAG's restrictions and recruited among youth as a means to get people without a compromised past, even if it meant sacrificing professional experience.

EDIT: There is one major exception for members of the former regime serving the MfS: Inoffizielle Mitarbeiter (Informal Collaborators or IMs). The MfS recruited IMs from Wehrmacht veterans, SS and NSDAP members, and the children of Nazi officials in cases when they were investigating groups they thought had ties to Western anti-communist organizations like the West Berlin-based Kampfgruppe gegen Unmenschlichkeit (Fighting Group Against Inhumanity). As with other IMs, the motivation for cooperation with the MfS is hard to parse out and varied by individual, but what the MfS wanted from these individuals was not their expertise, but information.

Sources

Bruce, Gary. "The prelude to nationwide surveillance in East Germany: Stasi operations and threat perceptions, 1945–1953." Journal of Cold War Studies. 2003;5(2):3-31.

_. The Firm The Inside Story of the Stasi. Oxford: Oxford University Press, 2010.

Gieseke, Jens. The History of the Stasi: East Germany's Secret Police, 1945-1990. New York: Berghahn Books, 2014.