Modified from an earlier answer
There were several mutually-reenforcing rationales behind the maintenance of the International Criminal Police Commission (ICPC) and INTERPOL. Firstly, the SS takeover of Interpol following the Anschluss represented another step gradual accumulation of administrative power for the SS's Reichssicherheitshauptamt (RSHA). The control of INTERPOL positioned the SS as a buffer block between foreign police and both German police and other bureaucratic agencies within Germany. Many of the preexisting concerns of INTERPOL, such as counterfeiting and passport forgeries overlapped with the RSHA's increasing assumption of control of the Jewish question within Germany. During the war, INTERPOL would be one of the means to maintain a liaison with and control foreign police when settling the Jewish question in occupied and Axis-allied states. As such, INTERPOL strengthened the SS's domestic position within the internal hierarchy of the Third Reich.
Yet there was also an important prestige element involved in the takeover of INTERPOL. The SS, and especially the young professionals of the RSHA, self-styled the organization as a modern bureaucratic entity that possessed increasing importance within the new order. Representation within the ICPC also potentially transferred to the RSHA a sheen of legitimacy through its participation within a global international organization. Heydrich had wanted to immediately move INTERPOL's headquarters to Berlin in 1938, but both Oskar Dressler, the longtime Secretary-General of INTERPOL, and Arthur Nebe persuaded him to keep the agency in Vienna to promote international police cooperation, albeit under the leadership of the Third Reich, and by extension, the RSHA. SS control of the ICPC meant the SS organization was not just a overgrown praetorian guard, but rather a professional group, at least in the minds of the SS leadership. This put the SS on par with other rising special law enforcement agencies, like the American FBI.
On a more sundry level, the SS used its contacts within the ICPC as a means both to derive intelligence and create a diplomatic backdoor channel to other states. The SS saw that continuing access to the ICPC files as boon to its own intelligence gathering operations and a way to make contacts to further the RSHA's increasing importance in directing espionage and counter-espionage operations. Finally, INTERPOL had a distinctly anti-communist bent and the RSHA saw participation within INTERPOL as part of a larger fight against Communism and the Comintern. Even prior to the Anschluss, INTERPOL fostered cooperation in attempting to apprehend Ernst Wollweber, whose Organisation Against Fascism and in Support of the USSR sabotaged the shipping of anti-Comintern countries in 1937. The RSHA leadership believed that it could push international police to cooperate in a united front against communism as it took over INTERPOL.
In practice, the coordination of INTERPOL to the Third Reich never quite worked out as the RSHA planned. Rather than legitimizing the SS, control of the ICPC made the SS look like a tool of German aggression and domination. Non-German police organizations increasingly saw the ICPC as a German-dominated organ that served German interests first. Even staunch anti-communists like J. Edgar Hoover and Belgium's Robert de Foy were reluctant to cooperate with the post-38 ICPC given the German domination of this international agency. During the war, INTERPOL became another tool for the SS to cajole and strong-arm various police forces in occupied Europe to cooperate with the various German occupation directives. The attempt to create a SS-dominated policing order ultimately fell flat between 1938-39 but the ICPC's preexisting networks between Germany and other European police organs were of reasonable utility during the war to further the SS's racial and political policies in occupied Europe.
Sources
Deflem, Mathieu. "The Logic of Nazification: The Case of the International Criminal Police Commission (" Interpol")." International Journal of Comparative Sociology 43, no. 1 (2002): 21-44.
Mazower, Mark. The Policing of Politics in the Twentieth Century: Historical Perspectives. Providence, RI [u.a.]: Berghahn Books, 1997.
Wildt, Michael. An Uncompromising Generation: The Nazi Leadership of the Reich Security Main Office. Madison, Wis: University of Wisconsin Press, 2009.