Assume we are given $N$ continuously or discretely observed paths solving a $d$-dimensional stochastic differential equation and admit mild assumptions that ensure the associated transition density exists. We propose constructing nonparametric, least-square projection estimators in dependence of the time $t$ and a space variable $x$ and calculate the estimator by minimizing a contrast over a product of finite dimensional spaces. Under our assumptions, we derive risk bounds for this estimator. We utilize these bounds to calculate the rate of convergence depending on the regularity of the density. We also propose a model selection procedure to derive estimators of the optimal dimensions of one or both orthonormal families generating the finite dimensional spaces of approximation. Finally, we illustrate the performance of our estimator for Ornstein-Uhlenbeck and Cox-Ingersoll-Ross processes in a small simulation study.