Responsibility for enforcing Prohibition fell to the Bureau of Prohibition at the Federal level, and to normal state and local agencies (e.g. state and local police forces). The FBI was frequently involved, as were various other Treasury and Justice Department agencies.
Note that the Bureau of Prohibition was in fact several different units over its lifespan, and used several slightly different names. It started out as merely a subunit of the IRS, transitioned to a full agency under the Dept of the Treasury, then ended up as part of the Justice Department and then a subunit of the FBI, before it lost its purpose in 1933 with the repeal of Prohibition.
The organization now lives on as the Bureau of Alcohol, Tobacco, Firearms, and Explosives (commonly known at the ATF), and independent agency within the Department of Justice.