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We study a branching Brownian motion Z in R^d , among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the process when hit by a particle. Considering a general offspring distribution, we find the decay rate of the annealed probability that none of the particles of Z hits a trap, asymptotically in time. This problem motivates us to find a general result about the speed of branching Brownian motion conditioned on non-extinction. In the proof, an appropriate skeleton decomposition for the underlying Galton-Watson process is provided and it is shown through a non-trivial comparison that the particles of finite line of descent do not contribute to the asymptotic decay rate of the annealed survival probability.