Bayesian optimization is a gold standard for query-efficient continuous optimization. However, its adoption for drug and antibody sequence design has been hindered by the discrete, high-dimensional nature of the decision variables. We develop a new approach (LaMBO) which jointly trains a denoising autoencoder with a discriminative multi-task Gaussian process head, enabling gradient-based optimization of multi-objective acquisition functions in the latent space of the autoencoder. These acquisition functions allow LaMBO to balance the explore-exploit trade-off over multiple design rounds, and to balance objective tradeoffs by optimizing sequences at many different points on the Pareto frontier. We evaluate LaMBO on a small-molecule task based on the ZINC dataset and introduce a new large-molecule task targeting fluorescent proteins. In our experiments, LaMBO outperforms genetic optimizers and does not require a large pretraining corpus, demonstrating that Bayesian optimization is practical and effective for biological sequence design.