Modified from an earlier answer of mine

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. The MfS made an explicit choice to avoid recruiting from those who had been members of the Third Reich's police and SS agencies. There was very few continuities in terms of personnel between the two dictatorships' security apparatus.

Unlike the SS's RSHA, the MfS did not recruit its initial members from existing police personnel or from young professionals. Although the idea of a "Red Gestapo" gained some currency during the Cold War with books like Bernhard Sagolla's Die Rote Gestapo (published in 1952 under the auspices of the Kampfgruppe gegen Unmenschlichkeit (fighting group against inhumanty) a West German anticommunist group with ties to both Bonn and the CIA), research by Gary Bruce and Jens Gieseke into MfS recruitment has shown that neither the Soviets nor the SED wanted members of the Third Reich's security and intelligence to gain access to the new state. 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 over the short term.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 mirror 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, the MfS and the SED's leadership expected former members of the NSDAP to work in tandem with Western intelligence agencies against the GDR. 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. This was one of the ways in which the MfS departed from the SD/RSHA. This SS organization wanted to establish itself quickly as a central police arm of the NSDAP state and thus it actively recruited among lawyers and other law enforcement professionals such Heinrich Müller or Arthur Nebe.

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 . 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 and social networks. Such IMs were insulated from the command and control structures of the MfS even if they were protected by the organization. These IMs received protection from prosecution, but in a pattern all too typical of the MfS, such protection carried with it a latent threat that it could be withdrawn if the IM was not cooperative.

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.