Alexander, a former Liberal Democrat MP who lost his seat in the 2015 General Election, was appointed vice president and corporate secretary by the bank's president Jin Liqun.
"I look forward to using my experience and skills, including as chief secretary to the treasury during one of the most challenging recent times for the UK economy, to help ensure the AIIB succeeds," Sir Alexander said.
It comes as Chancellor George Osborne confirmed that the UK intends to lead a constituency of the bank's members from wider Europe. Britain will have one of 12 seats on the AIIB's board of directors.
"Danny's appointment, together with the UK's position on the AIIB Board of Directors, demonstrates that the UK is playing a crucial role as the AIIB is formally established," Osborne said.
The AIIB has been touted as a potential rival to the International Monetary Fund and the World Bank, which critics say are dominated by the West. Britain raised eyebrows among the international community when it became the first major Western country to join the AIIB last year.