I would say that the Bank War (I'm assuming you're talking about Andrew Jackson and his "war" against the Bank of the United States) did not do anything to divide the North and the South. This was a battle that was waged on ideological and to some extent economic grounds, not sectional ones.
In general, Jackson had the support of most of the "common people" of the US (whether North or South) in his fight against the bank. It was a populist issue where you had the "regular" people lined up against the elites. Because there are "regular" people and "elites" in every section of the country this was not a sectional issue. Instead, it was one where the regular people supported Jackson and the more elite types supported the Bank.