No, it wasn't a crime. About 3.5 million GDR citizens fled, 3 million of those to West Germany (Source). There were state run refugee camps in the west where refugees got first orientation. A lot of people also had relatives in the west and could use them as a first station.

Regarding the question of whether those refugees faced legal problems in the west, they didn't. West Germany treated GDR citizens as own citizens. (Source: von Münch, Die deutsche Staatsangehörigkeit, de Gruyter, unfortunately in German)

So, they were citizens of West Germany. As such they had no legal problems whatsoever.

On a tangent, there was also the so-called "Alleinvertretungsanspruch", a legal doctrine upheld by the west until 1969 which stated that there was only one rightful German state in German territory and that the GDR was an unlawful military occupation. This was then changed slightly and delicately during Willy Brandt's chancellory and his new "Ostpolitik" where West Germany recognised the existence of the GDR for the first time, but without conceding points like automatic citizenship for GDR citizens.

I am sure it wasn't, however an interesting example or story would be nice.