Climate change is expected to increase the likelihood of drought events, with severe implications for food security. Unlike other natural disasters, droughts have a slow onset and depend on various external factors, making drought detection in climate data difficult. In contrast to existing works that rely on simple relative drought indices as ground-truth data, we build upon soil moisture index (SMI) obtained from a hydrological model. This index is directly related to insufficiently available water to vegetation. Given ERA5-Land climate input data of six months with land use information from MODIS satellite observation, we compare different models with and without sequential inductive bias in classifying droughts based on SMI. We use PR-AUC as the evaluation measure to account for the class imbalance and obtain promising results despite a challenging time-based split. We further show in an ablation study that the models retain their predictive capabilities given input data of coarser resolutions, as frequently encountered in climate models.