Underwater surveys provide long-term data for informing management strategies, monitoring coral reef health, and estimating blue carbon stocks. Advances in broad-scale survey methods, such as robotic underwater vehicles, have increased the range of marine surveys but generate large volumes of imagery requiring analysis. Computer vision methods such as semantic segmentation aid automated image analysis, but typically rely on fully supervised training with extensive labelled data. While ground truth label masks for tasks like street scene segmentation can be quickly and affordably generated by non-experts through crowdsourcing services like Amazon Mechanical Turk, ecology presents greater challenges. The complexity of underwater images, coupled with the specialist expertise needed to accurately identify species at the pixel level, makes this process costly, time-consuming, and heavily dependent on domain experts. In recent years, some works have performed automated analysis of underwater imagery, and a smaller number of studies have focused on weakly supervised approaches which aim to reduce the expert-provided labelled data required. This survey focuses on approaches which reduce dependency on human expert input, while reviewing the prior and related approaches to position these works in the wider field of underwater perception. Further, we offer an overview of coastal ecosystems and the challenges of underwater imagery. We provide background on weakly and self-supervised deep learning and integrate these elements into a taxonomy that centres on the intersection of underwater monitoring, computer vision, and deep learning, while motivating approaches for weakly supervised deep learning with reduced dependency on domain expert data annotations. Lastly, the survey examines available datasets and platforms, and identifies gaps, barriers, and opportunities for automating underwater surveys.