In this article we consider random walks in R^d such that the law of the increments has all exponential moments. For a large class of cones, we compute the exponential decay of the probability for such random walks to stay in the cone up to time n, as n goes to infinity. We show that the latter equals the global minimum, on the dual cone, of the Laplace transform of the random walk increments. Our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics.