Information in speech signals is not evenly distributed, making it an additional challenge for end-to-end (E2E) speech translation (ST) to learn to focus on informative features. In this paper, we propose adaptive feature selection (AFS) for encoder-decoder based E2E ST. We first pre-train an ASR encoder and apply AFS to dynamically estimate the importance of each encoded speech feature to SR. A ST encoder, stacked on top of the ASR encoder, then receives the filtered features from the (frozen) ASR encoder. We take L0DROP (Zhang et al., 2020) as the backbone for AFS, and adapt it to sparsify speech features with respect to both temporal and feature dimensions. Results on LibriSpeech En-Fr and MuST-C benchmarks show that AFS facilitates learning of ST by pruning out ~84% temporal features, yielding an average translation gain of ~1.3-1.6 BLEU and a decoding speedup of ~1.4x. In particular, AFS reduces the performance gap compared to the cascade baseline, and outperforms it on LibriSpeech En-Fr with a BLEU score of 18.56 (without data augmentation)