Timely outcome prediction is essential in healthcare to enable early detection and intervention of adverse events. However, in longitudinal follow-ups to patients' health status, cost-efficient acquisition of patient covariates is usually necessary due to the significant expense involved in screening and lab tests. To balance the timely and accurate outcome predictions with acquisition costs, an effective active sensing strategy is crucial. In this paper, we propose a novel risk-averse active sensing approach RAS that addresses the composite decision problem of when to conduct the acquisition and which measurements to make. Our approach decomposes the policy into two sub-policies: acquisition scheduler and feature selector, respectively. Moreover, we introduce a novel risk-aversion training strategy to focus on the underrepresented subgroup of high-risk patients for whom timely and accurate prediction of disease progression is of greater value. Our method outperforms baseline active sensing approaches in experiments with both synthetic and real-world datasets, and we illustrate the significance of our policy decomposition and the necessity of a risk-averse sensing policy through case studies.