Real-world user behaviors are often noisy due to factors such as human errors, uncertainty, and behavioral ambiguity, which can lead to degraded recommendation performance. To address this issue, recent approaches widely adopt self-supervised learning (SSL), particularly contrastive learning, by generating perturbed views of user interaction sequences and maximizing their mutual information to improve model robustness. However, these methods heavily rely on their pre-defined static augmentation strategies~(where the augmentation type remains fixed once chosen) to construct augmented views, leading to two critical challenges: (1) the optimal augmentation type can vary significantly across different scenarios; (2) inappropriate augmentations may even degrade recommendation performance, limiting the effectiveness of SSL. To overcome these limitations, we propose an adaptive augmentation framework. We first unify existing basic augmentation operations into a unified formulation via structured transformation matrices. Building on this formulation, we introduce AsarRec, an Adaptive Sequential Augmentation for Robust Sequential Recommendation. To enable stable end-to-end optimization of discrete and strongly constrained augmentations, AsarRec learns to generate transformation matrices by encoding user sequences into probabilistic transition matrices and projecting them into hard semi-doubly stochastic matrices via a differentiable Semi-Sinkhorn algorithm. To ensure that the learned augmentations benefit downstream performance, we jointly optimize three objectives: diversity (encouraging distinct views), semantic invariance (preserving semantic consistency among views), and informativeness (identifying augmentations most beneficial to recommendation). Extensive experiments on four benchmarks under varying noise levels validate the effectiveness of AsarRec, demonstrating its superior robustness and consistent improvements.