Vasiliki’s name day is January 1, the feast of Basil the Great.
Churches that follow the Old (Julian) Calendar keep the same feast on January 14 of the civil calendar. See everything commemorated on January 1.
The saint behind the name

Basil the Great
January 1, Great Feast, Βασίλειος
St. Basil the Great was born in Caesarea of Cappadocia in the fourth century, to a family of saints. Educated in Athens alongside his friend St. Gregory the Theologian, he set aside a brilliant secular career for the monastic life, and later served as Archbishop of Caesarea. He defended the divinity of Christ against Arianism, gave the Church the Divine Liturgy that bears his name, and organized one of the first great centers of Christian charity for the poor and the sick.
Name days: Vasilis, Vasileios, Vassilis, Vasile, Basil, Bill, Vasilios, Vassilios, Vasos, Vasily, Vasil, Billy, Vasiliki, Vassiliki, Vasilia, Vaso, Vasso, Vasia
Also celebrating on January 1
These names share the feast, many of them other forms of the same name:
Vasilis, Vasileios, Vassilis, Vasile, Basil, Bill, Vasilios, Vassilios, Vasos, Vasily, Vasil, Billy, Vassiliki, Vasilia, Vaso, Vasso, Vasia
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.