Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are more competitive than alternative techniques because the integrand in the frequency space often has a greater regularity than that in the physical space. However, when designing a numerical quadrature method for most Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: the choice of damping parameters to ensure integrability and control the regularity class of the integrand, and the effective treatment of high dimensionality. To address these challenges we propose an efficient numerical method for pricing European multi-asset options that is based on two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of damping parameters based on a proposed optimization rule. Then, we employ sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. The extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some Lévy models demonstrates the advantages of adaptivity and the damping rule on the numerical complexity of quadrature methods. Moreover, for the tested two-asset examples, the proposed approach outperforms the Fourier cosine expansion (COS) method in terms of computation time. Finally, we show a greater speed-up for up to six dimensions compared with the Monte Carlo method.