The Constitution creates the fundamental framework for the national government, which makes me wonder why the 18th Amendment ended up in the Constitution in the first place. Alcohol prohibition seems like a lower-level issue than those listed in the Constitution, and, for example, the Controlled Substances Act plays a similar role for a variety of other drugs and didn't require a constitutional amendment.
The United States Federal Government, under the Constitution, is one of limited, enumerated powers. Unlike the States which have a "general police power" to make all sorts of laws, the US Congress only has the power to legislate within the bounds of those particular powers, set forth in Article I Section 8 of the Constitution.
One of the enumerated powers is "To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes." This power has been interpreted by the United States Supreme Court to give Congress the power to regulate virtually any transaction that has an impact on interstate commerce--even to the extent of allowing federal regulation to reach wheat grown and used by a farmer on his farm because his doing so decreases the demand in interstate commerce in the case of Wickard v. Fillburn. The same logic allows the Controlled Substances Act to reach even purely intra-state drug transactions even when those transactions are legal under state law.
But that expansive interpretation of the Commerce Clause comes from the New Deal era, more than a decade after the adoption of the 18th Amendment. As of 1920, Congress's power was understood in a more limited way, and a specific constitutional amendment was required to grant Congress the power to prohibit alcohol completely.