Plasmonic sensors have been used for a wide-range of biological and chemical sensing applications. Emerging nano-fabrication techniques have enabled these sensors to be cost-effectively mass-manufactured onto various types of substrates. To accompany these advances, major improvements in sensor read-out devices must also be achieved to fully realize the broad impact of plasmonic nano-sensors. Here, we propose a machine learning framework which can be used to design low-cost and mobile multi-spectral plasmonic readers that do not use traditionally employed bulky and expensive stabilized light-sources or high-resolution spectrometers. By training a feature selection model over a large set of fabricated plasmonic nano-sensors, we select the optimal set of illumination light-emitting-diodes needed to create a minimum-error refractive index prediction model, which statistically takes into account the varied spectral responses and fabrication-induced variability of a given sensor design. This computational sensing approach was experimentally validated using a modular mobile plasmonic reader. We tested different plasmonic sensors with hexagonal and square periodicity nano-hole arrays, and revealed that the optimal illumination bands differ from those that are 'intuitively' selected based on the spectral features of the sensor, e.g., transmission peaks or valleys. This framework provides a universal tool for the plasmonics community to design low-cost and mobile multi-spectral readers, helping the translation of nano-sensing technologies to various emerging applications such as wearable sensing, personalized medicine, and point-of-care diagnostics. Beyond plasmonics, other types of sensors that operate based on spectral changes can broadly benefit from this approach, including e.g., aptamer-enabled nanoparticle assays and graphene-based sensors, among others.