Although over the past number of years there have been many advances in the materials aspects of topological insulators (TI), one of the ongoing challenges with these materials is the protection of them against aging. In particular, the recent development of low-carrier-density bulk-insulating Bi$_2$Se$_3$ thin films and their sensitivity to air demands reliable capping layers to stabilize their electronic properties. Here, we study the stability of the low-carrier-density Bi$_2$Se$_3$ thin films in air with and without various capping layers using DC and THz probes. Without any capping layers, the carrier density increases by ~150% over a week and by ~280% over 9 months. In situ-deposited Se and ex situ-deposited Poly(methyl methacrylate) (PMMA) suppresses the aging effect to ~27% and ~88% respectively over 9 months. The combination of effective capping layers and low-carrier-density TI films will open up new opportunities in topological insulators. ; Comment: 14 pages, 3 figures