Generally, it is considered OK if your nose is slightly open on a blow note, but it is not good to have your nose open while playing a draw note. It will make it difficult to learn certain techniques such as bending.
However, if you are playing a passage which has ten blow notes in a row, and then one draw note, it is okay to breathe through the nose while playing that draw note to get in some extra air.
Learning to not breathe through your nose will require you to consciously think about it while playing, until it becomes second nature. Are you comfortable breathing from your diaphragm?
It will really help to breath from your diaphragm, and not from your chest. So, make sure you are breathing properly.
Diaphragmatic breathing is explained in detail in our harmonica lessons: