Using environmental sensory data can enhance communications beam training and reduce its overhead compared to conventional methods. However, the availability of fresh sensory data during inference may be limited due to sensing constraints or sensor failures, necessitating a realistic model for multimodal sensing. This paper proposes a joint multimodal sensing and beam prediction framework that operates under a constraint on the average sensing rate, i.e., how often fresh sensory data should be obtained. The proposed method combines deep reinforcement learning, i.e., a deep Q-network (DQN), with a neural network (NN)-based beam predictor. The DQN determines the sensing decisions, while the NN predicts the best beam from the codebook. To capture the effect of limited fresh data during inference, the age of information (AoI) is incorporated into the training of both the DQN and the beam predictor. Lyapunov optimization is employed to design a reward function that enforces the average sensing constraint. Simulation results on a real-world dataset show that AoI-aware training improves top-1 and top-3 inference accuracy by 44.16% and 52.96%, respectively, under a strict sensing constraint. The performance gain, however, diminishes as the sensing constraint is relaxed.