Of course, every problem in P is decidable by definition of P.  This was mentioned in the previous answers.  

But there is another problem here that hasn't been addressed yet:  How do you present a class of graphs closed
under minors?  A priori, it is not clear that every class of graphs that is closed under minors (usually a class containing graphs of infinitely many isomorphism classes) has a reasonable representation as a finite object (that can be treated algorithmically) at all.
By finite representation I mean a formula that defines the class in one way or other or something similar.   

Now, the graph minor theorem gives us a nice representation of every such class:  Just list
the finite set of forbidden minors.  If this is your representation of the class, then you 
get your polynomial time algorithm that decides membership for the class.  

If you settle on another representation (and you have to come up with some uniform way to describe your class by finite objects to be able to say anything algorithmically at all, I would think), it might not be possible to come up with an algorithm that computes the finitely many forbidden minors from the representation of the class.