Based on the level of respect or formality, you can say "Which day of the week is best for you?" in Korean in 3 ways:

• Informal: museun yoil-i neo-ege jeil joa? (무슨 요일이 네게 제일 좋아?), museun yoil-i neo-ege jeil joeuni? (무슨 요일이 네게 제일 좋으니?)
• Formal: museun yoil-i dangsin-ege jeil joayo? (무슨 요일이 당신에게 제일 좋아요?), museun yoil-i dangsin-ege jeil jotseupnigga? (무슨 요일이 당신에게 제일 좋습니까?)
• Very Formal: museun yoil-i dangsin-gge jeil joeusipnigga? (무슨 요일이 당신께 제일 좋습니까?)

- which day: eoneu yoil (어느요일)
- week: ju (주)
- best: choisang-eui (최상의), jeil jotda (제일 좋다)
- you (indirect objective): neo-ege (너에게), dangsin-ege (당신에게), dangsin-gge(당신께)

Please refer to How to pronounce Korean Alphabet for the English pronunciation in this answer.