Do admissions care about one's minor?
Sure.  How much depends on the field and the department; some places have fairly strict requirements for what you have to have learned in undergrad, and if a minor can hit some of those things, then that works.  

However, in such cases, it tends to be less about the official minor and more about the coursework.  For example, if you're applying to grad school in a mathey field, getting good grades in linear algebra, differential equations, and real analysis will probably be a big help whether there's a math minor on your transcript or not.