Word for word, "How is your day (going)?" would be 오늘 어떻게 돼 가요? or 오늘 어떠세요/어때요?, but we rarely say such a phrase as a greeting.
(They are also slightly different in meaning as they're equivalent to "How's it going today (as compared with other days)?").
To my knowledge, there seems to be no general greeting in Korean that refers to how the day is turning out.
We say the usual "How are you?" type greetings without a reference to the way the day is going.
- 별일 없으세요? = Is everything going all right?
- 잘 지내시죠? = I hope you're doing fine.
- 잘돼 가세요? = Is everything going well? (when you know the person has something special going at the moment).