Pulsar candidate sifting is an essential process for discovering new pulsars. It aims to search for the most promising pulsar candidates from an all-sky survey, such as High Time Resolution Universe (HTRU), Green Bank Northern Celestial Cap (GBNCC), Five-hundred-meter Aperture Spherical radio Telescope (FAST), etc. Recently, machine learning (ML) is a hot topic in pulsar candidate sifting investigations. However, one typical challenge in ML for pulsar candidate sifting comes from the learning difficulty arising from the highly class-imbalance between the observation numbers of pulsars and non-pulsars. Therefore, this work proposes a novel framework for candidate sifting, named multi-input convolutional neural networks (MICNN). The MICNN is an architecture of deep learning with four diagnostic plots of a pulsar candidate as its inputs. To train our MICNN in a highly class-imbalanced dataset, a novel image augment technique, as well as a three-stage training strategy, is proposed. Experiments on observations from HTRU and GBNCC show the effectiveness and robustness of these proposed techniques. In the experiments on HTRU, our MICNN model achieves a recall of 0.962 and a precision rate of 0.967 even in a highly class-imbalanced test dataset.