To properly discuss Babylon would require an entire book. However, by the time Alexander the Great moved to make it his capital, it was already in decline.
Parts of the city that Alexander most certainly saw in his time are visible even without a risky trip to Iraq. Some of the murals from the chief palace are in the Metropolitan Museum of Art, while the Ishtar Gate is displayed in Berlin’s Pergamon Museum.
Unfortunately for Alexander, he didn’t have long to bask in his city after returning from his long foray into India. This coin was minted in 323 BCE in Babylon, which was where and when Alexander died. While it was probably minted before his death, since the mint mark would likely have changed soon after, we don’t really know.
During Alexander’s life, he gave the satrapy of Babylon to Mazaios, who had earlier opposed him under Darius II. When Mazaios died in 328 BCE, the satrapy was given to Stamenes. After Alexander’s death, Archon assumed the satrapy, until Perdikkas appointed Dokimos. Archon refused to give up the post, and he was killed in a battle with Dokimos.
He kept the satrapy for a short time, because during the Partition of Triparidisos it was given to Seleukos I Nikator. He had all of four years to enjoy the post, because in 317 BCE Antigonos I Monophthalmos chased him out and appointed Aspesias as his own governor.
In 312 BCE, Seleukos returned by Babylon and retook his satrapy. The city remained under Seleukid control until most of the inhabitants were moved to the new city of Seleukeia in 275 BCE, and Babylon lost its role as an important player in world history.