“Άγιος Βασίλης” and “Αϊ-Βασίλης” are both correct.
“Άγιος Βασίλης” is more formal. “Αϊ-Βασίλης” is a form that we encounter most often in Christmas songs and tales for children.
“Άγιος” is an adjective and it is always conjugated when it is used:
ο Άγιος Βασίλης (Nominative)
του Άγιου Βασίλη (Genitive)
τον Άγιο Βασίλη (Accusative)
“Αϊ” is not conjugated, it doesn't take an intonation mark and connects with a dash with the following word. In grammar it is called “προτακτικό”.
ο Αϊ-Βασίλης (Nominative)
του Αϊ-Βασίλη (Genitive)
τον Αϊ-Βασίλη (Accusative)