We are not aware of any halachic restriction on cord blood banking, except for limitations on collection on Shabbat and Yom Tov. Cord blood banking is widely practiced here in Israel in hospitals that cater to religious populations.
It should be noted that there are two forms of cord blood banking – public banking (where cord blood is donated freely for use by anyone who needs it, similar to a blood bank) and private banking (where one pays money to save the cord blood for private use). The first is encouraged by health organizations. The latter is frowned upon by the American Academy of Pediatrics as it has not been shown to be of benefit unless there is a currently ill child in the family or the family is prone to specific genetic diseases. While these considerations are not strictly halachic, they have some impact on Jewish medical ethics.