It's an interesting question, in spirit at least, although you could make it more precise by saying exactly *which* algebraic-style axioms you're talking about---there are several proposals.  

Here's something between a comment and an answer.  To avoid the question of which axiom system we're using, I'll just talk about 2-categories (*weak* ones, i.e. bicategories).  2-categories are defined in such a way that "all diagrams commute", and the same goes for n-categories in general.  Sure, there's a finite axiomatization, but we only know it's the right one because it allows us to prove that "all" diagrams commute.

You can write a computer program that spits out, in turn, all the diagrams that are supposed to commute.  In that sense, the axioms are recursively enumerable. 

An interesting observation is that 2-categories can be *finitely* axiomatized.  In fact, this is the axiomatization that everyone meets: you have a pentagon, and a triangle, and some naturality squares.  But in principle you have an *infinite* collection of coherence axioms: the "all diagrams" I referred to.  

So the theory of 2-categories *is* finitely axiomatizable, but I don't know of any explanation of why it *had* to be.  More generally, if you take a finitely axiomatizable  algebraic theory (e.g. monoids) and categorify it (obtaining e.g. monoidal categories), I don't know whether that categorified theory must inevitably be finitely axiomatizable.

(Of course, finite axiomatizability is not the same as computability, but it's a closely related question.)