We investigate theoretically Bose-Einstein condensation of an ideal, trapped Bose gas in the presence of Rashba spin-orbit coupling. Analytic results for the critical temperature and condensate fraction are derived based on a semiclassical approximation to the single-particle-energy spectrum and density of states and are compared with exact results obtained by explicitly summing discrete energy levels for a small number of particles. We find a significant decrease of the critical temperature and of the condensate fraction due to finite spin-orbit coupling. For a large coupling strength and a finite number of particles N, the critical temperature scales as N 2/5 and N 2/3 in three and two dimensions, respectively, contrasted to the predictions of N 1/3 and N 1/2 in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.