We prove upper bounds on the one-arm exponent η1 for dependent percolation models; while our main interest is level set percolation of smooth Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson-Voronoi and Poisson-Boolean percolation. More precisely, in dimension d = 2 we prove η1 ≤ 1/3 for Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in general dimensions we prove η1 ≤ d/3 for finite-range fields and η1 ≤ d − 2 for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. We also establish a new Russo-type inequality for smooth Gaussian fields which we use to prove the sharpness of the phase transition for finite-range fields.