We investigate the feasibility of distributed acoustic sensing (DAS) for marine traffic monitoring in the Trondheimsfjord, Norway. We deployed a DAS system on a fiber-optic telecommunication cable trenched at the seabed of the fjord and recorded continuous acoustic signals generated by passing vessels. In this study, we investigate if an alternative method and workflow can be used to monitor marine traffic despite the noisy environment and poor coupling of the cable to the surrounding seabed sediments. We apply 2D image processing techniques and various signal processing filters to the data to achieve further noise reduction. We analyzed the data using a method named persistent homology to detect direct arrivals of vessel signals. Inverting for the traveltimes of the detected P-wave arrivals enables us to localize the acoustic source. Our results show that persistent homology can be an effective tool for analyzing continuous signals. It is a promising data processing technology for detecting and tracking vessels in coastal areas. We compared our localization results for a vessel to the GPS locations logged by the automatic identification system (AIS) and found good agreement. Furthermore, we demonstrate the ability of the method to localize successfully a previously undetected vessel within reasonable positioning errors, that was not sending an AIS signal. These findings demonstrate the potential of our presented method which has significant implications for maritime security and environmental protection. While other research studies have focused on localizing vessels and marine mammals in calm and deep water, we present an approach that allows localizing vessels (and potentially other objects) in noisy, shallow waters. Overall, our study highlights the valuable capabilities of DAS for marine traffic monitoring but underscores the need for continued research to fully realize its potential in this area.