The reinstitution of conscription was an incredibly contentious issue for the Adenauer government and the Ohne Mich (without me) campaign was incredibly strong within the FRG. Protests against the Bonn regime in Berlin could play into the hands of the GDR as it tended to portray any protest against Bonn as a sign that Germans had rejected the FRG. The result of this was there were a number of ways to evade conscription built into the system.
One of the more common ways to evade military or state service was for young German men to go set up residence within West Berlin. West Berlin was an anomalous entity within the FRG in that it was never a fully recognized component of the state, but still under the administration of Bonn. This odd status was the byproduct the Allied wartime negotiations over the city which held that Berlin's final status would be decided by a formal peace treaty between Germany and the Allied powers. The emerging Cold War had pushed such a final settlement into the seemingly distant future and West Berlin's administration took on a permanent provisional character. Since Berlin was a hot-button issue for most of the early Cold War, men from West Berlin were exempt from conscription when the Adenauer government reinstituted it in 1957. West Berlin's anomalous status not only made instituting conscription there legally questionable, but it also threatened the political stability of this de facto FRG exclave.
Exemption from conscription in West Berlin became a major issue during the 1960s as the FRG student movement connected service in the Bundeswehr with the wider struggle in Vietnam. Many of the 68ers' leadership came from the West Berlin milieu and the Freie Universität Berlin (FU), an academic institution ironically set with significant American support in 1948. But it was not just conscription that made many young West Germans gravitate to the city. The decline of West Berlin's industry and flight from the Cold War meant that many buildings and flats were available for prices much lower than their equivalents in the FRG proper, so that a band of relatively poor youth could live like princes in the city, or even set up shop in an abandoned building. Institutions like the FU also provided a pretext for residence to the city. Additionally, the culture of West Berlin proved very attractive to those whose political convictions meshed with a desire not to serve in the Bundeswehr. FU's student culture was often very transgressive and the university's administrative structure was much more open than the more traditional and stuffy German university system. The 68ers also championed a more liberal vision of sexual politics that was at odds with the conservative mores of the Adenauer generation. Both draft-dodging and this wider intellectual climate of rebellion created a feedback loop of sorts by the late 1960s and early 70s in which both strengthened each other.
Bonn's response to draft-evasion in West Berlin was two-fold. Firstly, it tightened up its regulations and control of access to the city. This meant that anyone caught in the city without having the proper justification for being there would be subject to FRG laws for draft evasion. This move to tighten access to the city worked in conjunction with other police actions against squatters in West Berlin and restore order to a city that had become one of the epicenters for student violence and protests. In addition to this stick, Bonn also used the carrot of expanding the conscientious objector (CO) provisions for military service. The conservative CDU had pushed through a very narrow interpretation of CO in the Bundestag in which an applicant had to prove that their objection was of a fundamental nature rather than one determined by his situation. Such a narrow reading of CO status favored religious over secular justifications because they were much easier for the state to evaluate. CO's could apply for an Ersatzdienst (substitute service) where they would perform a term of non-military service.
The SPD governments of the 1970s lowered the threshold for CO status by eliminating some the inquisitional barriers for CO status and streamlining the process to only be a written declaration. Bonn also expanded the Ersatzdienst for official COs and renamed it Zivildienst (these have no real English translation and the German government terms them "alternative civil service", but in German, "Zivil" has much more culturally elevated connotations than "Ersatz"). Civil service helped alleviate draft evasion because it gave FRG men much more options than service in the Bundeswehr, thus acting as a pressure valve of sorts. Additionally, as the passions of the 68ers fizzled out in the 1970s, military service became a much less contentious issue in German politics than it was in the late 1960s. CO applications though continued to go on an upwards trend. While they hovered in the low four digits during the Adenauer period, they grew to nearly 12000 in 1969 and averaged in the 40-50000 range during the 1970s and 80s.
Overall, draft evasion for the Bundeswehr did occur, but not on the scale of other conscript armies in the Cold War. Although some cultural conservatives and former veterans wrung their hands over young Germans' lack of enthusiasm to serve, Bonn was restrained in prosecuting these cases. Neither the Adenauer and Erhard chancellorships sought to make this a major political issue, especially as it gave both the GDR as well as the SPD ammunition against the conservative government. Moreover, conscription was also in line with the mainstream ideological precepts of the SPD that saw a people's army as more congruent with democratic values than a professional force. Draft evasion was possible, but other semi-legal means were also available to West German men that did not want to serve.
Sources
Large, David Clay. Germans to the Front: West German Rearmament in the Adenauer Era. Chapel Hill: University of North Carolina Press, 1996.
Longhurst, Kerry. "Why aren't the Germans debating the draft? path dependency and the persistence of conscription." German Politics 12, no. 2 (2003): 147-165.
Von Bredow, Wilfried. "Conscription, Conscientious Objection, and Civic Service: The Military Institutions and Political Culture of Germany, 1945 to Present." JPMS: Journal of Political and Military Sociology 20, no. 2 (1992): 289.