I can't speak to current anti-Semitism in Spain, but I can describe how it developed in the 1920's and 1930's. At that time there were few jews in Spain--a legacy of the expulsion of the jews in the 15th century. So the fact that anti-Semitism was an important part of the Spanish Civil War is more than a bit odd.

The reason that this happened was that many Spaniards seized upon The Protocols of the Elders of Zion in a uniquely Spanish way. Led by a Father Tusquets--a Spanish priest and writer--many Spanish writers believed that there was indeed an international jewish conspiracy to destroy western civilization, including Spain and the Catholic Church. Fr. Tusquets' writing was widely distributed and became popular. He describes a "Judeo-Masonic-Bolshevik conspiracy" (or a "Judeo-Muhammadean-Masonic-Bolshevik conspiracy") to destroy Spain and the Catholic Church. While he did raid Masonic temples for documents to 'prove' his allegations, his entire premise was fiction. However, the fears of the right and the middle class in Spain were such that this fiction was accepted as fact. Tusquets was an effective propagandist and appropriated the rhetoric of the left into his work, giving it a sinister cast. Alerted to Tusquets' book, the Vatican (specifically Cardinal Pacelli, who later became Pope Pius XII) stripped it of it's nihil obstat--a sign that the Spanish Catholic Church was going down a path that the Vatican did not approve of.

By the time the Spanish Civil War began many figures on the right were convinced that jews were indeed setting out to destroy their very way of life. Many of the executions of leftists in Nationalist territory were explained by the allegation that the victim was a jew or jew sympathizer. These executions of noncombatants numbered in the tens of thousands--many of them with the stated motivation of killing jews that weren't even present in the country to begin with. Leftist anticlerical violence didn't help the perception of leftists as desirous of the destruction of all the right held dear, which reinforced the fears of those on the right.

I'm less of an expert of what happened after the Spanish Civil War. For his part, Fr. Tusquets turned down the offers of high level posts in Franco's regime and went into education, as well as repudiating his book. How sincere his turnaround was is up for debate, but the fact that he did not accept positions of great power and influence after the war speaks to his apparent contrition. The damage was already done, though. Tens of thousands were dead, many of them because the fiction of a jewish conspiracy to destroy Spain was believed. I can't assert that the current situation in Spain is a result of the writings of Tusquets and others, but anti-Semitism in the 1930's was widespread among the right largely due to their influence.

For further reading, Paul Preston's The Spanish Holocaust does a great job on this topic. Jose M. Sanchez also addresses it in his The Spanish Civil War as a Religious Tragedy.