We study the typical behavior of the harmonic measure in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index α ∈ (1, 2]. Let µ n denote the hitting distribution of height n by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation n. We extend the results of [11] to prove that, with high probability, the mass of the harmonic measure µ n carried by a random vertex uniformly chosen from height n is approximately equal to n −λα , where the constant λ α > 1 α−1 depends only on the index α. In the analogous continuous model, this constant λ α turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for λ α , we are able to show that λ α decreases with respect to α ∈ (1, 2], and it goes to infinity at the same speed as (α − 1) −2 when α approaches 1. Keywords. size-biased Galton-Watson tree, reduced tree, harmonic measure, uniform measure, simple random walk and Brownian motion on trees.