I've read about the eighteenth amendment, but I can't imagine what would push prohibition past simple federal law and into constitutional amendment. It just seems a bit severe.
There is an earlier AskHistorians discussion of why prohibition in the U.S. was instituted via a constitutional amendment.
From a comment on that thread by /u/polo_commando:
Congress needs specific authorization in the Constitution to do anything. There is a line that "Congress shall have the power to regulate commerce among the several states..." which, in today's constitutional jurisprudence, has come to the point of meaning "anything related to the economy or behavior can be regulated by Congress." The majority of laws and regulations in force today, including drug prohibitions, are through this authority.
Back in the 1920's, the Supreme Court had not yet given such an expansive interpretation to that passage--it still meant "Congress can regulate trade that crosses state lines", more or less. So, in order to get authorization to make laws prohibiting alcohol, they had to change the Constitution itself via an amendment.
And from a comment on that thread by /u/quellthesparkle:
But to that point in history, no amendment had ever been overturned, so the proponents of prohibition wanted it to be a permanent prohibition of alcohol instead of just implementing it through a law which could be overturned.