Since the Babylonian Exile occurred centuries before the Christian Church was established, it had no effect on the power and the prestige of the church.
The Babylonian Exile actually enhanced the power and prestige of the Jewish priesthood. As there was no longer a king, the priests became the outright leaders of Jewish society, establishing a theocracy that remained even after the Return from Exile.
There was no weakening; as the old testament was first written down in Babylon, the Abrahim religion and its descendants now had concrete texts.