Accurate estimation of biological brain age from three dimensional (3D) T$_1$-weighted magnetic resonance imaging (MRI) is a critical imaging biomarker for identifying accelerated aging associated with neurodegenerative diseases. Effective brain age prediction necessitates training 3D models to leverage comprehensive insights from volumetric MRI scans, thereby fully capturing spatial anatomical context. However, optimizing deep 3D models remains challenging due to problems such as vanishing gradients. Furthermore, brain structural patterns differ significantly between sexes, which impacts aging trajectories and vulnerability to neurodegenerative diseases, thereby making sex classification crucial for enhancing the accuracy and generalizability of predictive models. To address these challenges, we propose a Deeply Supervised Multitask Autoencoder (DSMT-AE) framework for brain age estimation. DSMT-AE employs deep supervision, which involves applying supervisory signals at intermediate layers during training, to stabilize model optimization, and multitask learning to enhance feature representation. Specifically, our framework simultaneously optimizes brain age prediction alongside auxiliary tasks of sex classification and image reconstruction, thus effectively capturing anatomical and demographic variability to improve prediction accuracy. We extensively evaluate DSMT-AE on the Open Brain Health Benchmark (OpenBHB) dataset, the largest multisite neuroimaging cohort combining ten publicly available datasets. The results demonstrate that DSMT-AE achieves state-of-the-art performance and robustness across age and sex subgroups. Additionally, our ablation study confirms that each proposed component substantially contributes to the improved predictive accuracy and robustness of the overall architecture.