Yes, they are both correct. The meaning is the same, but the context and assumptions are different.

1. 'Did you see Mary today?'

This is a past simple construction referring to a finished - 'closed' - period of time. You might say this in the evening or at the end of the day. For example, if Mary is a classmate of yours, someone might say 'Did you see Mary today?' after school has finished. The assumption is that you no longer have an opportunity to see Mary today.

2. 'Have you seen Mary today?'

This is a present perfect construction referring to an unfinished - or 'open' - period of time. You could ask this at any time of the day, and the assumption is that you still have an opportunity to see Mary today. Even if you haven't seen her yet, it is still possible that you will see her later on today. The possibility of seeing Mary today still exists.