  We consider the Steklov eigenvalue problem on a compact pinched negatively
curved manifold $M$ of dimension at least three with totally geodesic
boundaries. We obtain a geometric lower bound for the first nonzero Steklov
eigenvalue in terms of the total volume of $M$ and the volume of its boundary.
We provide examples illustrating the necessity of these geometric quantities in
the lower bound. Our result can be seen as a counterpart of the lower bound for
the first nonzero Laplace eigenvalue on closed pinched negatively curved
manifolds of dimension at least three proved by Schoen in 1982.
  The proof is composed of certain key elements. We provide a uniform lower
bound for the first eigenvalue of the Steklov-Dirichlet problem on a
neighborhood of the boundary of $M$ and show that it provides an obstruction to
having a small first nonzero Steklov eigenvalue. As another key element of the
proof, we give a tubular neighborhood theorem for totally geodesic
hypersurfaces in a pinched negatively curved manifold. We give an explicit
dependence for the width function in terms of the volume of the boundary and
the pinching constant.
