Hand-intensive manufacturing processes, such as composite layup and textile draping, require significant human dexterity to accommodate task complexity. These strenuous hand motions often lead to musculoskeletal disorders and rehabilitation surgeries. We develop a data-driven ergonomic risk assessment system with a special focus on hand and finger activity to better identify and address ergonomic issues related to hand-intensive manufacturing processes. The system comprises a multi-modal sensor testbed to collect and synchronize operator upper body pose, hand pose and applied forces; a Biometric Assessment of Complete Hand (BACH) formulation to measure high-fidelity hand and finger risks; and industry-standard risk scores associated with upper body posture, RULA, and hand activity, HAL. Our findings demonstrate that BACH captures injurious activity with a higher granularity in comparison to the existing metrics. Machine learning models are also used to automate RULA and HAL scoring, and generalize well to unseen participants. Our assessment system, therefore, provides ergonomic interpretability of the manufacturing processes studied, and could be used to mitigate risks through minor workplace optimization and posture corrections.