There are some misconceptions in this question here.

The erection of the Berlin Wall was in no small measure the end result of East Germans fleeing to West Germany (FRG). The position of the FRG was that all Germans could claim FRG citizenship and that Bonn was the temporary, legitimate government of Germany until a final peace treaty was signed between the German government and the Allies. This meant that any GDR citizen who made it to FRG territory could apply for citizenship in the FRG.

The problem was getting to the FRG and the Cold War made it incredibly difficult. The inner German border was already starting to be patrolled by military forces as early as 1947. Both the GDR and the Soviets turned the inner German border into a strip of mines, guard towers, and bunkers by the 1950s. To make matters worse, the FRG and NATO also armed their half of the border. While Germans could and did walk across the inner German border in the 1940s, such activities had become decidedly dangerous by the early decade of the Cold War.

This left West Berlin as the easiest FRG territory for a refugee to access. GDR rail-lines and other transport networks did run through the divided city. Some refugees could buy tickets and flee during the trains' stopover in Berlin. Additionally, the large expansive nature of Berlin's urban geography made it easy to walk into West Berlin. There were a good deal of so-called "Sachsenganger", migrant GDR workers whose jobs were in West Berlin but were residents of the GDR, prior to 1961. Most refugees that reached West Berlin applied for asylum citizenship and were flown out of the city via the established air corridors. The erection of the Wall ended this migrant labor and also the last "easy' egress for an aspiring refugee. The Wall crudely and at great costs to prestige and lives did resolve the refugee crisis.

Nonetheless, the Wall still was the best option for fleeing the GDR. The trickle of post-61 refugees would try both avenues, but Berlin remained disproportionately popular area for escape. The Wall was certainly a deadly strip of territory, but it was still nowhere near as militarized as the inner German border. This slightly hyperbolic US training video form the mid-Cold War illustrates some of the features of this area. Even though it was officially an "anti-fascist defensive rampart," the Wall was a tad less daunting than the other border available for refugees. The Western Berlin brigades were not much of a threat to the GDR and the Wall was less militarized than the inner border. The latter's fortifications were not only meant to stop refugees, but also to prevent a NATO drive into the GDR.