This is an argument that can go either way. Roman civilization itself has had a profound influence in shaping the later course of European history, and this applies to both the Republic and the Empire.
I think there are several arguments that can be used in favor of the Roman Republic. For one thing, there is the Roman Republic's influence on the later Roman Empire, given that the Empire itself emerged out of the Late Republic's instability in the civil wars. In addition, however, solely on its own merits, you can argue for the Republic's long-term political influence as one of the ancient world's key examples of representative government, an example that would do much to shape the rise of modern democracy as it is seen in the world today.
On the other hand, if we look at the height of Roman wealth and power, we would be looking at the Roman Empire (which would have exerted a greater hold on European imaginations moving into the Middle Ages and beyond). More important, however, might be the rise of Christianity within the Roman world. Even after Rome's collapse, Christianity would remain the most powerful source of shared unity in the post-Roman world, exerting a profound influence in shaping the further evolution of European civilization through to the modern era. In many respects, the rise of Christianity might well have been the most profound and long-reaching of all of Rome's legacies, and this would belong entirely to the Empire.