In this paper, an efficient approach for the computation of the fair value of a basket option as well as its Greeks is presented. Both European and American options are considered; the determination of the price of the former leads to the solution of a parabolic partial differential equation, whereas for the latter a variational inequality needs to be solved. The variational formulations are discretized in terms of finite differences in time and finite elements in space. By enforcing the inequality constraints of the American option via Lagrange multipliers, the discrete inequality can be reformulated as a set of semismooth equations that is solved in terms of a primal-dual active set strategy. In order to estimate the Greeks, we construct a piecewise multilinear interpolant of the pricing function with respect to the coefficients of the differential equation. The partial derivatives of this interpolant serve as an approximation of the Greeks that can thus be evaluated for any combination of assets and market parameters. The number of function evaluations necessary for the interpolation is reduced by using dimension-adaptive sparse grids for the discretization of the parameter space. Several numerical examples illustrate the robustness and applicability of the schemes.