Advanced computer vision technology can provide near real-time home monitoring to support "aging in place" by detecting falls and symptoms related to seizures and stroke. Affordable webcams, together with cloud computing services (to run machine learning algorithms), can potentially bring significant social benefits. However, it has not been deployed in practice because of privacy concerns. In this paper, we propose a strategy that uses homomorphic encryption to resolve this dilemma, which guarantees information confidentiality while retaining action detection. Our protocol for secure inference can distinguish falls from activities of daily living with 86.21% sensitivity and 99.14% specificity, with an average inference latency of 1.2 seconds and 2.4 seconds on real-world test datasets using small and large neural nets, respectively. We show that our method enables a 613x speedup over the latency-optimized LoLa and achieves an average of 3.1x throughput increase in secure inference compared to the throughput-optimized nGraph-HE2.