What were the reasons that the Prohibition in the United States was passed as a constitutional amendment, rather than a normal federal law, or a set of state laws?
Prohibition required a constitutional amendment, because the Federal government does not have the power to regulate intra-state commerce.
The majority of states and many localities had already banned the sale of alcohol. The progressive and women's suffrage movements saw banning alcohol as a way to improve living conditions for women and children and reduce the power of saloon-centric machine politics.
I believe this was to make the local State laws, that were a hodgepodge of enactments on a State level, more uniform and to bring the enforcement into the Federal realm. This was especially easy since the Temperance Movements had become politically powerful in the early 20th Century and in the election of 1917 the pro-Prohibitionists finally had a majority in Congress and moved things along. The actual text is:
Without the Volstead Act though the Amendment was very vague, the Volstead Act was more important as it was the one that set down enforcement of the ban. Note the text says nothing about consumption.
At the time the Constitution WAS seen as the enshrinement of laws, getting an Amendment gave weight to your cause and Federal enforcement. I don't have official sources on this but there are arguments that prior to the use of the Commerce Clause to give Federal enforcement powers an Amendment was the way to get National enforcement. Although looking more at varous sources the most common rationale given for a national-law is the Temperance Movement already had many States enacting Prohibitive Laws, but this would make it Federal and not allow Liquor interests to interfere locally and influence local politicians.
Prohibition existed in many localities around the country prior to the amendment, but the answer to your question seems to be because the powers that wished to enact prohibition had overwhelming public support. So, the group that wanted prohibition did it in part because they could.
To further flesh out some points of MichaelF's answer, prohibition was already starting to take hold around the country. According to Irving Fisher (a professor of economics who testified during congressional hearings on prohibition in April of 1926) in his book Prohibition at its Worst:
Professor Fisher notes that prohibition was essentially forced upon the large cities of the East Coast by the rest of the country. The Anti-Saloon League is cited as a particularly effective organization that saw prohibition through to its ultimate success.
It is also interesting to note, as the Howard McBain (a law professor at Columbia University) did in 1928, the sheer amount of popular support for prohibition:
Interestingly, the Prohibition Amendment invalidated all state and local regulations that dealt with prohibition because the Constitution became the supreme law in this area, and granted police power to the feds to enforce the amendment.
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As other answers have mentioned, there were state and local laws that prohibited alcohol before the Constitutional amendment. And there is the obvious fact that a Constitutional amendment is a more permanent measure than a normal law, which would require a more complex measure to overturn. (There may be a parallel to the moves in recent years to enshrine gay marriage bans in state Constitutions; in most cases there were already laws in place, but a Constitutional amendment was thought of a something that might be "more permanent" and wouldn't be overturned easily if a different party came into power.)
However, the question is a bit more complicated, because one might argue that -- at the time -- a "normal federal law" couldn't be used. Or, if it were used, it could not achieve the sweeping goals of the temperance movement in truly prohibiting alcohol. Furthermore, a law prohibiting alcohol at that time might have been successfully challenged and overturned by the Supreme Court.
That's the short explanation. To understand the situation in detail, we have to consider the history of Constitutional law in the United States.
The Constitution contains a set of enumerated powers for the federal government. Originally, Congress had no power to pass laws that could not derive directly from one of these powers; any other government powers were granted to the states (as reinforced by the Tenth Amendment). However, the exact limitations on these powers were disputed almost immediately as new legal situations arose. The 19th century saw a number of these enumerated powers become slightly expanded or curtailed in various Supreme Court cases.
Nevertheless, the presumption in the early 20th century was that Congress could not create laws which went beyond those original enumerated powers. The Supreme Court frequently struck down federal laws that could not be related back to the Constitution (perhaps most famously in the string of court decisions that repeatedly invalidated much of Franklin Roosevelt's "New Deal" in the early to mid 1930s on the grounds that the federal government did not have those powers). While the Supreme Court jurisprudence of this era tended to focus on economic freedom issues (the so-called Lochner era), any legislation that took power away from states or individuals could be suspect if it was not in the enumerated powers.
The regulation of "interstate commerce" is contained in the enumerated powers, but it was still largely interpreted as an interstate issue, i.e., the federal government could legally become involved mostly when merchandise or businesses went across state lines. Thus, early attempts at drug restriction by the federal government tended to focus on taxes for items going across state lines, rather than outright bans or detailed regulation.
An interesting case study here is the Harrison Narcotics Tax Act, which was passed in 1914 to regulate opiates and cocaine. The act did not make such drugs illegal, but merely required special registrations and taxes on those who manufactured or distributed them. The tax on physicians and medical use was low, but for non-medical use, the tax was so exorbitant as to effectively prohibit distribution or sale of the drugs. This second tax was challenged in court, where it was claimed that the act was not a tax -- which would be a Constitutionally enumerated power -- but rather a federal infringement of the police power of the states, which of course was its intent if not its literal mechanism. Nevertheless the Supreme Court upheld the act in United States v. Doremus (1919) while reaffirming that the federal government could not usurp the police power of the states. But the Supreme Court did later object to use of the act to arrest doctors who chose to prescribe these drugs without sufficient medical justification, e.g., just to drug addicts. As Justice McReynolds wrote in Linder v. United States (1925), "Obviously direct control of medical practice in the states is beyond the power of the federal government." This shows the kind of complex debates about precisely how far federal power could go, which a Constitutional amendment would avoid at this time.
Another option was for the federal government to pass uniform acts, such as the Uniform State Narcotic Drug Act of 1934. A "uniform act" is a collaborative measure intended to be adopted by individual states to ensure a uniform process governing a particular issue. However, states cannot be forced to adopt such regulations; initially only 9 states adopted this early narcotics act, and a public relations campaign was launched by FDR and others to convince the public of "reefer madness" and thus adopt narcotics legislation.
Since alcohol regulation is not specified in the Constitution, Congress had no explicit power to regulate it. And, as affirmed for example by the Supreme Court in the Matter of Heff (1905):
Nonetheless, Congress was making progress using its enumerated powers leading up to full Prohibition. In the Wilson Act (1890), through its power to regulate the postal system, Congress affirmed states' ability to regulate mailed liquor once it entered a state. Regulation of the postal system and interstate commerce led to further restrictions on imported liquor in "dry" states or communities, including the C.O.D. Act (1909), the Webb-Kenyon Act (1913), and ultimately the Reed Amendment (1917). Many such laws faced challenges in the Supreme Court and were only upheld when justified in a particular enumerated power.
Given that much alcohol was locally brewed or distilled in the early 1900s, regulation of "interstate commerce" would not have been sufficient to prohibit alcohol completely, and a "uniform act" was essentially a state-by-state approach which was already underway.
Thus, a Constitutional amendment was (at that time) the only way to create a nationwide law actually banning alcohol sale and distribution in general.
The Lochner era came to an end with the so-called "Switch in Time that Saved Nine," when Justice Owen Roberts started voting to uphold sweeping federal powers in 1937, thereby creating a 5-4 majority on the Supreme Court that was reluctant to declare legislation unconstitutional on the basis of the enumerated powers. (Whether Roberts's decision had anything to do with FDR's threat to enlarge the Supreme Court and appoint enough new justices to override his opponents is something that historians have long debated. Roberts himself declared that his decision to "switch sides" had nothing to do with FDR's plan.)
In any case, laws regulating controlled substances rapidly followed the 1937 shift, such as the 1938 Federal Food, Drug, and Cosmetic Act and various new laws regulating marijuana, etc. By the early 1940s, there were effectively few restrictions left that constrained the "interstate commerce" clause, which was now interpreted to be able to regulate any economic activity, from commerce only within states to personal goods consumed on one's own property.
Thus, Congress prior to 1937 had only limited powers to regulate manufacture, sale, and transportation of most substances (alcohol included), and mostly resorted to taxation as a remedy. After about 1942 Congress has had basically unlimited power to pass such regulations as long as they do not interfere with specific individual rights. Thus, a sweeping prohibition of alcohol from the 1910s-1930s required a Constitutional amendment, but subsequent prohibition and regulation of various other drugs has been achieved through "normal federal laws."
Federal laws are based on powers granted to the Senate and the House of Representatives that are enumerated in the U.S. Constitution. It took the 18th Amendment to the Constitution for Congress to be empowered to prohibit the production and sale of alcohol.
The resulting federal enabling legislation was known as the Volstead Act.