First of all, Lysenko did not oppose genetics in general, he was against so-called "formal genetics". He was the director of the Institute of Genetics of the Academy of Sciences of the USSR since 1940.
In 1939 he said in a speech:
Unfairly comrades Mendelists allege that we profess closing of Genetics (...) Genetics is necessary and we struggle for its development and flourishing
He authored a number of works on genetics, including
"About two directions in genetics" (1939, conference theses)
"Agrobiology. Works on questions about genetics, selection and seed breeding" (1949)
Article "Genetics" for Encyclopedy of Agriculture and for Great Soviet Encyclopedy (1946)
"To the question about old and new genetics" (1947)
Odessa Institute of Selection and Genetics was named after Lysenko in 1948.
As you know in 1933 the USSR experienced a heavy hunger, or at least the Soviet leadership thought it was hunger. It prompted them to search an effective method to quickly increase productivity of agriculture.
There were several reasons why it was Lysenko who enjoyed the government's support.
Lysenko promised quick, immediate results, while traditional genetics at the time did not bring practical results at all. Lysenko pointed that even if he had not a complete theory, it was the practical results that matters. At the time among Soviet scientists there was a widespread belief that many spheres of bourgeois science are not practical, pure scholastic and drain resources. To achieve practical results Lysenko used a number of methods of various effectiveness, such as vernalization, nested planting and others. Some of his methods such as "plant chasing" proved effective and used to this day in Tajikistan and Uzbekistan.
Lysenko developed an image of a native ethnic Russian peasant "just from a plow" who learned himself, an image closely associated with Lomonosov, while many of his opponents (including Vavilov) were ethnic Jews whom Stalin did not trust. Vavilov in addition was of bourgeois origin. Many also thought that Jews have no natural capabilities and scent in the sphere of agriculture, which ethnic Russian peasants have by nature.
By the Lysenko's followers it was claimed that traditional genetics gives scientific support to racism, eugenics and social inequality. This was a sound argument in the face of the advancing Nazism. Some felt that even if traditional theory correct, the danger of it being applied to humans makes it worth to oppose it by any means.