When did babylon begin to dominate Mesopotamia

In the 18th Century BC, during the reign of Hammurabi.

Explanation:

During the 2nd millennium B.C. Amorites and Syrians settled down in Mesopotamia, which is again divided into several kingdoms.  However, in the 18th century B.C. Hammurabi, king of Babylon, established a unique and stable state that stretched from the Persian Gulf to the city of Mari.  City of Babylon became a very important cultural and religious center of Mesopotamia.