Deep learning is rapidly becoming a ubiquitous signal-processing tool in big-data experiments. Here, we present the results of a proof-of-concept experiment which demonstrates that deep learning can successfully be used for production-scale classification of compact star clusters detected in HST UV-optical imaging of nearby spiral galaxies (D 20 Mpc) in the PHANGS-HST survey. Given the relatively small and unbalanced nature of existing, human-labelled star cluster datasets, we transfer the knowledge of state-of-the-art neural network models for real-object recognition to classify star clusters candidates into four morphological classes. We show that human classification is at the 66% : 37% : 40% : 61% agreement level for the four classes considered. On the other hand, our findings indicate that deep learning algorithms achieve 76% : 63% : 59% : 70% for a star cluster sample within 4Mpc ≤ D ≤ 10Mpc. We further tested the robustness of our deep learning algorithms to generalize to different cluster images. For this experiment we used the first data obtained by PHANGS-HST of NGC1559, which is more distant at D = 19Mpc, and found that deep learning produces classification accuracies 73% : 42% : 52% : 67%. We furnish evidence for the robustness of these analyses by using two different state-of-the-art neural network models for image classification, which were trained multiple times from the ground up to assess the variance and stability of our results. Through ablation studies, we quantified the importance of the NUV, U, B, V and I images for morphological classification with our deep learning models, and find that, as expected, the V-band is the key contributor as human classifications are based on images taken in that filter. The methods introduced in this article lay the foundations to automate classification for these objects at scale, and motivate the creation of a standardized star cluster classification dataset, developed and agreed upon by a range of experts in the field.