In this paper, we consider a class of stochastic differential equations driven by symmetric non-degenerate $α$-stable processes (including cylindrical ones) with $α\in (1,2)$. We first establish a quantitative estimate for the Euler scheme under bounded drift $b(x)$, with an explicit dependence on $ \| b \|_{L^\infty}$. Then we obtain the weak convergence rates for the case where the drift coefficient belongs to a Besov space of negative order.