Humans rely on ankle torque to maintain standing balance, particularly in the presence of small to moderate perturbations. Reductions in maximum torque (MT) production and maximum rate of torque development (MRTD) occur at the ankle with age, diminishing stability. Ankle exoskeletons are powered orthotic devices that may assist older adults by compensating for reduced torque and power production capabilities. They may also be able to assist with ankle strategies used for balance. However, their effect on standing balance in older adults is not well understood. Here, we model the effects ankle exoskeletons have on stability in physics-based models of healthy young and old adults, focusing on the potential to mitigate age-related deficits in MT and MRTD. Using backward reachability, a mathematical technique for analyzing the behavior of dynamical systems, we compute the set of stable center of mass positions and velocities for sex and age adjusted nonlinear models of human standing balance with an ankle exoskeleton. We show that an ankle exoskeleton moderately reduces feasible stability boundaries in users who have full ankle strength. For individuals with age-related deficits, there is a trade-off. While exoskeletons augment stability at low center of mass velocities, they reduce stability in some high velocity conditions. Our results suggest that ankle exoskeletons using established control strategies might have unforeseen negative effects on stability, especially for individuals who are most likely to benefit from them.