Facing the COVID-19 pandemic, governments have implemented a wide range of policies to contain the spread of the virus. During the pandemic, large amounts of COVID-19-related tweets emerge every day. Real-time processing of daily tweets may offer insights for monitoring public opinion about intervention measures implemented. In this work, lockdown policy in New York State has been set as a target of public opinion research. This task includes two stages, stance detection and opinion monitoring. For the stance detection stage, we explored several combinations of different text representations and classification algorithms, finding that the combination of Long Short-Term Memory (LSTM) with Global Vectors for Word Representation (GloVe) outperforms others. Due to the shortage of labeled data, we adopted the data distillation method for the training data augmentation. The augmentation of the training data allows to improve the performance of the model with a very small amount of manually-labeled data. After applying the distillation method, the accuracy of the model has been significantly improved. Utilizing the enhanced model, automatically classified tweets are analyzed over time to monitor the public opinion. By exploring the tweets in New York from January 22nd until September 30th, 2020, we show the correlation of public opinion with COVID-19 cases and mortality data, and the effect of government responses on the opinion shift. These results demonstrate the capability of the presented method to effectively and efficiently monitor public opinion during a pandemic.