We propose a multifidelity dimension reduction method to identify a low-dimensional structure present in many engineering models. The structure of interest arises when functions vary primarily on a low-dimensional subspace of the high-dimensional input space, while varying little along the complementary directions. Our approach builds on the gradient-based methodology of active subspaces, and exploits models of different fidelities to reduce the cost of performing dimension reduction through the computation of the active subspace matrix. We provide a non-asymptotic analysis of the number of gradient evaluations sufficient to achieve a prescribed error in the active subspace matrix, both in expectation and with high probability. We show that the sample complexity depends on a notion of intrinsic dimension of the problem, which can be much smaller than the dimension of the input space. We illustrate the benefits of such a multifidelity dimension reduction approach using numerical experiments with input spaces of up to three thousand dimensions.