ATHENS—Evangelia Koskina just received her first pay increase in six years, but she worries that it could be bad news for the café where she works.

Greece, like fellow euro member Spain, decided this year to sharply increase the minimum wage, despite the legacy of high unemployment left by Europe’s long debt crisis. In both countries, left-leaning governments are betting—against economic orthodoxy—that such increases will be good for growth and employment, as well as for their voter support.

The...