St. Joseph’s intercession against demons is not well-documented but has been largely been proven by experience. For example, in the early 20th-century book Go to Joseph, Our Unfailing Protector, the author gives a brief summary of how missionaries invoked his intercession to drive out demonic influences. Sacred writers agree in saying that the intercession of St. Joseph is an efficacious means of overthrowing the tyranny of Satan and propagating the kingdom of Jesus Christ. Indeed this fact is borne out by numerous evidences a result of which is that the holy Patriarch has come to be invoked by Holy Church with the title terror of demons. Missionaries in [pagan] countries are unanimous in acknowledging the help they have derived from St. Joseph to that effect.
One may also wonder why St. Joseph would be given such a title. In the late 1930s, Pope Pius XI began to see the serious threat that communism posed to the common good and called upon St. Joseph to protect the Church from the errors of communism. As a result, Catholics fervently began to pray to St. Joseph, specifically under the title “Terror of Demons” to combat the atheism and the atheistic ideas of communism.
St. Joseph’s most singular mission entrusted to him by God was the caring for and protecting of the Holy Family. Joseph was endowed with graces enough to strike fear into the forces of darkness. The demons don’t stand a chance against St. Joseph. St. Faustina writes: “Saint Joseph urged me to have a constant devotion to him. ... He has promised me this special help and protection (1203).” aleteia.org ; saginaw.org