We associate $p$-adic Galois representations to globally generic cusp forms on $GSp(4)$, over a totally real field, with a Steinberg component at some finite place. At places $v$ not dividing $p$ one has local-global compatibility, the local correspondence being that defined by Gan and Takeda. In particular, the rank of the monodromy operator at such a place $v$ is determined by the level of the $v$-component of the cusp form. Moreover, the Swan conductor is essentially the depth.