I am a citizen from Latin America, and many of the recent dictatorships here tried to justify themselves by using the "fight against communism" as an excuse. Are there any other countries(outside LA) that founded an authoritarian state using the same objective? And are there countries that, facing the same "problem", didn't need to estabilish some kind of dictadorship or authoritarian state to deal with it? (Sorry for eventual grammar mistakes)
The Spanish Civil War resulted in Franco's dictatorship, and the Nationalists justified their cause in large part as being anticommunist--before, during, and after the war. The fact that the communists were a tiny part of the leftist electoral victories in 1931 and 1936 didn't bother the right-wing anticommunist rhetoricians. Further, there was the invention of a Jewish-Masonic-Bolshevik plot (and sometimes it went a step further, to be a Jewish-Muhammadean-Masonic-Bolshevik plot) to destroy Spain and the Catholic Church (and by extension, civilization). After WWII, Franco's long-time anticommunist stance would make him useful to the US.
For two views of anticommunist theory in Spain, Paul Preston's The Spanish Holocaust and Jose M. Sanchez's The Spanish Civil War as a Religious Tragedy reach the same conclusions while approaching the subject from two different starting points (Preston is more sympathetic to the Republicans while not letting his bias unduly affect his work, and Sanchez is a Catholic scholar who has no problem criticizing other Catholics).
As /u/Domini_canes already pointed out, the primary characteristic of Francoist Spain was the anti-Communism of the various factions. This is what led to limited Spanish involvement in World War II exclusively on the Eastern Front, and what allowed Franco to maneuver the country into the United States' good graces with the rise of the Cold War.
For a more in depth look at Francoist Spain, I'd direct you to this post I did on their involvement in WWII and this one that deals more with the Cold War (and don't miss Domini's follow up either.)
Instances of dictatorships in the name of anti-communism that we often forget in the western world are those of two consecutive leaders in Cold War South Korea: Syngman Rhee from 1948-1960, and Park Chung-hee who ruled from 1962 until he was assassinated in 1979. Both Presidencies were characterized by multiple human rights abuses and hardliner conservative policies.
Rhee came into power after being appointed with US support, and then questionably "elected" soon after. His rule had elements of authoritarian rule in his governing stemming from social conflict in the Korean War, and was eventually forced to resign in the student-led April Revolution of 1960. Little economic gain was made for the whole of South Korea under his rule.
Park Chung-hee was an army officer who quickly rose through the ranks of his newly-formed Democratic Republican Party following Rhee's resignation. He was an elected leader, but his party was largely nationalistic and authoritarian. He is often credited by older Koreans as the father of modern Korea because of his economic modernization, but Korea's economy began to do poorly in the 70's, as he was assassinated by his director of the KCIA(Korean Central intelligence Agency) in 1979. By this time, he had much less support.
Examples aside, US backed anti-communist dictatorships were often less interested in fighting communism as they were staying rich & powerful, but as long as they retained an official stance of anti-communism the American government turned a blind eye to human rights abuses and gave the dictatorships full support in the name of "democracy"
Sources:
http://news.bbc.co.uk/onthisday/hi/dates/stories/october/26/newsid_2478000/2478353.stm#startcontent
http://legacy.wilsoncenter.org/coldwarfiles/index-33794.html
http://www.britannica.com/EBchecked/topic/444035/Park-Chung-Hee
http://asianhistory.about.com/od/southkorea/p/Park-Chung-Hee-Biography.htm
http://countrystudies.us/south-korea/11.htm
EDIT: Park Chung-hee formally came into power in 1962 as acting president, being elected in 1963. This is contrary to my previous information, saying he came into power in 1960.
Greece comes to mind. The communist resistance in WWII fought a civil war immediately following the defeat of the Germans with the right wing government which was backed by the British and later the Americans. The right wing parties won the 1946 elections, which were boycotted by the communists, leaving the country divided for decades. Eventually a military anti communist junta was established in 1967 after a coup d'etat under the pretext of ending political intability and making sure the communists would't gain political power. Source for the period just following the war: Savage Continent by Keith Lowe.