If you want an abstract criterion, it is known that a finitely generated profinite group is uniquely determined by its finite quotients (this is Theorem 3.2.9 of [Ribes Zalesskii][1]: if $G_1$ is a finitely generated profinite group, and $G_2$ is a profinite group with the same finite quotients, then $G_1\cong G_2$). Of course, if $G_1$ has rank $r$, then all finite quotients of $G_1$ will have rank $r$.

Conversely, suppose $G$ is a profinite group such that every quotient has rank $\leq r$ (and some quotient of rank $r$). By a diagonalization argument, we may find $r$ elements of $G$ such that the image in any finite quotient of $r$ is a set of generators. Thus, these elements form a topological generating set for $G$. 

Thus, $rank(G)$ is the supremum of ranks of finite quotients of $G$. 



  [1]: http://books.google.com/books?id=47ouE_XSJZYC&lpg=PP1&dq=ribes%2520zalesskii&pg=PP1#v=onepage&q&f=false