The origins of the term "antisemitism" and also anti-Semite, anti-Semitic, etc. go back to the late 18th century German writer August Ludwig von Schlözer who coined the precedent term "Semitismus". While this term may originally have been neutral, by the mid-19th century it had taken on negative connotations, basically meaning those traits and stereotypes which were believed to pertain to the Jews.
The first record use of the term "anti-semitismus" was in 1860, by the Jewish writer Moritz Steinschneider. He used the term to describe Ernst Renans views of the Jews. The term is also recorded in the Staats-Lexicon of Karl von Rotteck.
However the person who is usuually credited with inventing the term is Wilhelm Marr. Marr used the term as a self-definition for his political views, and his organization, the Antisemiten-Liga. Marr introduced the word and his organization in his 1879 pamphlet "The Victory of Jewr over Germandom", which was the first known publication in German to advance the idea of a worldwide Jewish conspiracy.
The term "Antisemitism" was adopted by a number of like minded groups in Germany and France in the 1880s. There was a French newspaper L'Antisemitique by 1883, a German periodical Antisemitische-correspondenz and even a political party - Antisemitische Volkspartie founded in 1889 that used the term as a political self-identification.(Antisemitism; a historical encyclopedia of prejudice and persecution Richad S. Levy ed. Santa Barbara, CA; ABC-CLIO
pp.22, 24-5)
Some examples of the terms use
https://upload.wikimedia.org/wikipedia/commons/f/f2/1889_French_election_poster_for_antisemitic_candidate_Adolphe_Willette.jpg
http://archive.org/details/lesjuifscontrela00drumuoft