The talk is based on joint result with Yuliya Mishura (Kiev University) and Enkelejd Hashorva (Lausanne University). We study probabilities $P(\forall t \in \mathbb{T}, X_f + f(t) \le u(t))$ of non-crossing a boundary $u$ by a centered Gaussian process $X$, indexed by a separable metric space $\mathbb{T}$, with drift $f$. We establish upper and lower bounds for non-crossing probabilities, which are matching in the sense that they determine precise logarithmic asymptotics for probabilities in the case of large drift.