Woodrow Wilson was in his second term when Prohibition was enacted. He was more of a peripheral figure in the story of Prohibition, someone acted upon rather than acting directly.
In the 1916 election, neither presidential candidate paid much attention to the matter of Prohibition, even though powerful organizations like the Anti-Saloon League and the Women's Christian Temperance Union were campaigning hard for "dry" Congressional candidates.
The president saw alcohol use as a "social and moral" issue (source) and didn't think it was a good idea to have the federal government regulate it. He enjoyed the occasional drink as much as the next guy and thought that it was okay to encourage temperance, but not to totally prohibit the sale of liquor (source).
Still, 17 of the 30 states that Wilson won were dry states, something that the "drys" wouldn't let him forget after he was elected. In January of 1916, Congress passed the Sheppard Act, which banned alcohol in the District of Columbia. Wilson was in a pickle. He knew that whether he signed or vetoed the bill, he'd lose some supporters. Not wanting to create divisiveness in his party, he signed it. Plus, Wilson had another matter on his mind: World War.
Throughout 1916 and 1917, war was tearing Europe to shreds, and Wilson was persuading Congress to pass spending bills to provide weapons for U.S ships to defend themselves against German submarines. The Sheppard Act was a local D.C. matter, and Wilson probably didn't want to risk losing Congressional support for what he saw as more important defense issues that were surely going to arise in the immediate future. In other words, choose your battles.
The U.S. entered WWI on April 6, 1917. Advocates of alcohol bans thought that a nation of drunks wouldn't be up to the task of providing a strong and reliable defense industry. Besides, all that grain could be better put to use feeding the the country.
On the other hand, labor leaders like Samuel Gompers tried to persuade the prez not to take away the working man's beer while rich folks were happily partying with their stash of expensive liquor. Wilson was stuck in the middle again. Congress tried to amend a bill regulating the wartime use of food to include a ban on using grain to make beer and liquor. Wilson struggled for a compromise. In the end, he signed a bill prohibiting the use of grain for making hard liquor, but allowed the continued production of wine and beer.
The Prohibition train had left the station, however, and in December 1917, Congress passed the Prohibition Amendment and sent it to the states. By January 1920, the 18th Amendment had been ratified and Congress went to work passing legislation that would define the terms used in the Amendment and detail how it would be enforced.
Wilson had hoped that the law would be more or less the same as the wartime compromise, but the eventual National Prohibition Act (aka the Volstead Act) turned out to be a lot more limiting. Wilson vetoed it, but his veto was immediately overridden. Prohibition was the law of the land.
Wilson had lost this battle, but he won the war—literally. For his role in brokering the Treaty of Versailles that ended WWI, he was awarded the Nobel Peace Prize in 1919. The U.S. emerged from the war as a major world power.
And there was the matter of another Amendment: the 19th. Thanks to Wilson's support, women gained the right to vote in 1920.
All in all, a pretty decent record for old #28.