This paper provides a catalogue of stars, quasars, and galaxies for the Southern Photometric Local Universe Survey Data Release 2 (S-PLUS DR2) in the Stripe 82 region. We show that a 12-band filter system (5 Sloan-like and 7 narrow bands) allows better performance for object classification than the usual analysis based solely on broad bands (regardless of infrared information). Moreover, we show that our classification is robust against missing values. Using spectroscopically confirmed sources retrieved from the Sloan Digital Sky Survey DR16 and DR14Q, we train a random forest classifier with the 12 S-PLUS magnitudes + 4 morphological features. A second random forest classifier is trained with the addition of the W1 (3.4 $mu$m) and W2 (4.6 $mu$m) magnitudes from the Wide-field Infrared Survey Explorer (WISE). Forty-four percent of our catalogue have WISE counterparts and are provided with classification from both models. We achieve 95.76% (52.47%) of quasar purity, 95.88% (92.24%) of quasar completeness, 99.44% (98.17%) of star purity, 98.22% (78.56%) of star completeness, 98.04% (81.39%) of galaxy purity, and 98.8% (85.37%) of galaxy completeness for the first (second) classifier, for which the metrics were calculated on objects with (without) WISE counterpart. A total of 2,926,787 objects that are not in our spectroscopic sample were labelled, obtaining 335,956 quasars, 1,347,340 stars, and 1,243,391 galaxies. From those, 7.4%, 76.0%, and 58.4% were classified with probabilities above 80%. The catalogue with classification and probabilities for Stripe 82 S-PLUS DR2 is available for download.
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