This paper deals with Audio-Visual Speech Recognition (AVSR) under multimodal input corruption situations where audio inputs and visual inputs are both corrupted, which is not well addressed in previous research directions. Previous studies have focused on how to complement the corrupted audio inputs with the clean visual inputs with the assumption of the availability of clean visual inputs. However, in real life, clean visual inputs are not always accessible and can even be corrupted by occluded lip regions or noises. Thus, we firstly analyze that the previous AVSR models are not indeed robust to the corruption of multimodal input streams, the audio and the visual inputs, compared to uni-modal models. Then, we design multimodal input corruption modeling to develop robust AVSR models. Lastly, we propose a novel AVSR framework, namely Audio-Visual Reliability Scoring module (AV-RelScore), that is robust to the corrupted multimodal inputs. The AV-RelScore can determine which input modal stream is reliable or not for the prediction and also can exploit the more reliable streams in prediction. The effectiveness of the proposed method is evaluated with comprehensive experiments on popular benchmark databases, LRS2 and LRS3. We also show that the reliability scores obtained by AV-RelScore well reflect the degree of corruption and make the proposed model focus on the reliable multimodal representations.