The Garnier system is an extension of the sixth Painlev\'e equation from a viewpoint of the isomonodromy deformation of a Fuchsian system. Its $q$-difference analogue was proposed by Sakai as the connection preserving deformation of a linear $q$-difference system. Recently, we formulated the $q$-Garnier system in a framework of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1$. On the other hand, the $q$-Garnier system admits a particular solution in terms of the basic hypergeometric series ${}_{n+1}\phi_n$. Then it becomes the next problem to investigate an action of the extended affine Weyl group on ${}_{n+1}\phi_n$. In this talk, we give an answer to this problem. Namely, we give a left action of a subgroup of the extended affine Weyl group on a vector whose components are described in terms of ${}_{n+1}\phi_n$. Hence $q$-contiguity relations and a linear $q$-difference equation for ${}_{n+1}\phi_n$ can be derived from the extended affine Weyl group systematically.