Well to start off the LDS church really didn't have a long tradition against alcohol. Joseph Smith received the original revelation in 1833, commonly called The Word of Wisdom. In it God specifically said it was only a recommendation, and not a commandment. Likewise it only banned hard liquors. Wines and mild barley drinks (aka beer) were still God approved.

And that's largely how it was treated for decades. Smith himself owned and operated a tavern. He also had wine and tobacco with his last meal before he was killed.

His successor Brigham Young owned and operated a distillery, several bars, and in fact was probably the largest alcohol distributor in the Utah territory. He made a particular whiskey sold under the label of "Valley Tan." The bottles tended to be heavily reused. Eventually "valley tan" became the name of the nastiest style of home-brewed moonshine in the mountain west as moonshiners loved to reused Young's bottles.

That's not to say that no one ever tried to get the Mormons on board with the Word of Wisdom. There are several sermons were Young asked the members to treat it as if it were a commandment. But he tended to focus on tobacco. Apparently he saw that as a drain on the Utah economy. They could produce booze locally, but most tobacco had to be imported and this was a source of cash flowing out of the territory. It's actually interesting to read his conference talks on the subject. As he over time switched from "no tobacco," to "no tobacco in church," to can you at least chaw discretely during services. With Young's death even these desultory efforts ceased for a while.

It wasn't until 1898 that the church leadership started to look at the Word of Wisdom again. Even then emphasis was on the rule that church members should refrain from eating meat. There was even a row between youth groups and the LDS church when the church was asked to close the saloon they owned at the Saltaire resort. The church refused to close the bars that they owned.

This all changed when Heber Grant became church president in 1902. The church did close the saloons they owned (not just at Saltaire) and Grant made a point of emphasizing the old largely ignored dietary code. In 1905 water replaced wine as the blood of Christ in the sacrament. That wasn't a completely new change. Water had long been acceptable, but the change made it mandatory.

And the right to go to the temple started to be attached to dietary code compliance. This was initially a soft prohibition and older members who liked a glass before bed were tolerated. Only "drunkards" were initially banned. And it wasn't until 1933 that temple attendance required complete absetence from alcohol.

Likewise passage of local prohibition happened fairly late. Utah only became a dry state in 1917, three years before national prohibition passed.

Most of this so far has been taken from Thomas Alexander's article From Principle to Requirement from the Dialog magazine. From this point on I'll move to Helen Papanikolas' Bootlegging in Zion: Making and Selling the 'Good Stuff' and try to answer your actual question.

Like a lot of the country a lot of counties and towns prohibited alcohol before statewide or national bans were enacted. This actually set the stage for prohibition because there were already communities of moonshiners and bootleggers. So when prohibition fell there was already a population used to making and discretely selling booze. In all, from 1925 to 1932 federal agents in Utah seized over 400 distilleries, 25,000 gallons of spirits, 8,000 gallons of malt liquors, 13,000 gallons of wine, and 332,000 gallons of mash. All too often police officers and other local officials were implicated in the busts. And in 1923 the Utah Attorney General commented that drinking was just as bad as before prohibition.

Interestingly when repeal came, Utah was the state that cast the deciding vote to end prohibition. When the sale of alcohol was renewed the first two legal stores sold $54,866 worth of booze in their first two weeks of operation.

All in all Utah's Prohibition experience largely mirrored that of other states.