(meaning specifically the ruins pictured here)
Was it during the rule of the Seleucids c. 304 BCE - 64 BCE or was it earlier during 585 BCE - 485 BCE with Nebuchadnezzar II and/or the Achaemenid Empire?
edit: Populated enough for it to be considered a city.
Babylon was inhabited throughout the Hellenistic period, and survived into the Arsacid, Parthian, and Aramaic periods. In fact, the city did not fall into larger disuse until after the Muslim conquests of the Middle East in the 7th c. CE. Of course, the city underwent numerous changes over the centuries, and many of the buildings in the map you linked were not used after the first millennium BCE. But the city itself remained vibrant for several centuries after the periods you mentioned.