Nucleic acid binding proteins (NABPs) including DNA-binding proteins (DBPs) and RNA-binding proteins (RBPs), play vital roles in gene expression. Accurate identification of them is crucial important. In previous studies, there are two existing challenges: one is the problem of ignoring DNA- and RNA-binding proteins (DRBPs), the other is cross-predicting problem referring to DBP predictors predict DBPs as RBPs, and vice versa. In this study, we proposed a computational predictor, called DeepMC-iNABP, aiming at solve these difficulties by utilizing multi-class classification strategy and deep learning approaches. DBPs, RBPs, DRBPs and non-NABPs as separate classes data were used for training the DeepMC-iNABP model. The results on test data collected in this study and two independent test datasets showed that DeepMC-iNABP has strong advantage in identifying the DRBPs and has ability of alleviating the cross-prediction problem to a certain extent.