JOEL - Mormons are so influential because the majority of people(75%) living in the state are Mormons. If most of the voters are Mormon then what they believe(like not drinking alcohol) is going to be represented in what they vote for. The same kind of thing goes on in other states for other things(abortion, same-sex marriage, death penalty, etc.). For example, the Baptists are considered to be highly responsible for "dry" counties in the deep south.
Many laws and administrative actions are based upon the moral standards of a society. If most citizens believe that it is wrong to kill or steal or lie, our laws will include punishment for those acts. If they believe that it is right to care for the poor and needy, our laws will facilitate those activities. Society continually legislates morality. So it is in Utah, and if most are Mormons the legislation will reflect Mormon values. You need to remember some of the important instances in which the efforts of churches and clergy in the political arena have influenced American public policies. For example, the slavery controversy was seen as a great moral issue and became the major political issue of the nineteenth century because of the preaching of clergy and the political action of churches. A century later, churches played an important role in the civil rights movement, and, ten years later, clergymen and churches of various denominations were an influential part of the antiwar movement that contributed to the end of the war in Vietnam.
Officially the LDS church teaches that civic laws should not interfere with religious practices, nor should religious institutions manipulate governments to their advantage. The Church has issued the following statement regarding the Church and Politics:

"The Church does not endorse political candidates or parties in elections. The Church does not advise its members how to vote. Church members should study the issues and candidates carefully and prayerfully and then vote for those they believe will act with integrity and will most nearly carry out their ideas of good government. Members are encouraged to participate as responsible citizens in supporting measures that strengthen society morally, economically, and culturally. They are urged to be actively engaged in worthy causes to improve their communities and make them more wholesome places in which to live and rear families. Political candidates should not imply that their candidacy is endorsed by the Church or its leaders." (May 6, 1996)

The LDS Church is not trying to monopolize the State Government, but it just so happens that many of the citizens involved in Utah state government are members of the Church who are trying to serve citizens, the majority of which are mostly members of the Church; so of course that will have some influence on policies and laws.
As a matter of prudence, our church has confined its own political participation within a far smaller range than is required by the law or the Constitution. Other churches have chosen to assert the full latitude of their constitutional privileges and, in the opinion of some, have even exceeded them.
The Church never takes any official stand on any political issue except for the few that directly affect the moral health of the community or country.(ie. abortion, same-sex marriage, etc.)
I haven't lived in Utah for twenty years so you will have to ask someone there about the high liquor tax. If you don't like it, drive to Evanston, Wyoming; you can get it much cheaper there :-)