"Sylvester was instrumental in stemming the spread of Arianism throughout the Western church"

I'm curious; could you expand upon his methodology in achieving this goal?

While it's certainly true that Sylvester's efforts were not completely effective in removing Arianism from the Western church - Alaric and his posse are generally believed to have been at least semi-Arians when they sacked Rome 80-odd years after Sylvester's death - he was an active supporter of the homousion cause and supported the bishops/metropolitans who stood behind the Cappadocians in this effort at Nicea I. While Sylvester was not present in person, Rome's position was a boon for the Homousion cause at the Council. He also sought to only allow orthodox bishops to be consecrated, but this control seems to have been strongest in southern Europe. Thus, the holdout of Arianism in the north eventually died without political/theological support.

During the later part of his life, Sylvester was a supporter/mentor of Marcus and Julius, both of whom succeeded Sylvester as pontiff. Marcus' pontificate was short, some seven months, but Julius' was longer, running until 352. His greatest contribution against Arianism was his support of Athanasius: The Egyptian bishops had set up an Arian bishop, Pistus, whom Julius refused to recognize in the wake of the Synod of Rome, 340-341. Although Constantine II allowed some of the semi-Arian/Arian bishops to retain their sees willy-nilly, Julius' support of Athanasius made it clear that the theological leadership in the West was opposed to any manifestations of Arianism.