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We put this question to Shaykh Muhammad ibn Saalih al-‘Uthaymeen, may Allaah preserve him, who replied as follows:
This is haraam. It is not permissible to establish this kind of bank, because this is human milk, and the milk from different mothers will be mixed, so that no one will know who is the mother. In Islam, drinking the milk of a woman creates the same relationship as does a close tie by blood (i.e., it has an effect on whom one may and may not marry, etc.). If the milk is of any kind other than human, then there is nothing wrong with milk banks.
And Allâh knows best.