How relevant is a candidate's BA institution when applying to graduate schools in the US?
The general answer is 


  Very relevant, but considered in the context of the rest of the application.  


It seems hard to say more in the absence of a more focused question.  Before you do that, you might try to search the site: I can't say that all possible versions of this kind of question have already been asked and answered here....but certainly many of them have.