The aging of the human adaptive immune system takes and interesting and distinct form when compared with the aging of other organs and processes. There is accumulating cellular damage, yes, but there is also a very important and ongoing process of misconfiguration: an otherwise largely sound system ceasing to function correctly not because it is damaged per se, but because it becomes poorly organized. That is, potentially, a much easier problem to solve than many of the other challenges presented to us by aging.
The way in which this misconfiguration happens is described back in the Fight Aging! archives. The short version is that the adaptive immune system remembers all threats, but has a limit to the number of cells it can produce; eventually too much of its quota is taken up by memory cells and too little by cells that are actually equipped to destroy things. A small range of persistent but otherwise mostly harmless viruses, such as cytomegalovirus (CMV), greatly speed up this process by hanging around for decades and constantly provoking the immune system into uselessly devoting ever more memory cells to their existence.
There are a range of possible ways to deal with this issue, with varying levels of complexity, cost, and permanence: adding more cells cultured from the patient's own stem cells, destroying the unwanted memory cells using targeted therapies of the sort under development in the cancer research community, and so forth. One of the presentations given at the SENS5 conference discusses the latest research in this area:
Aging is associated with an increased susceptibility of older individuals to new and emerging infections; poor responses to vaccination compound this vulnerability. ... In both mouse and man, repeated interactions between reactivating viruses such as CMV and antiviral T cells leads to memory T-cell inflation (MI) with increasing accumulation of these cells over the lifespan. It was hypothesized that MI may exact a price for the immune system: competition between inflating, CMV-specific memory T-cells and naive T-cells supposed to defend against all other infections may impair the maintenance of a diverse naive T-cell pool, consequently leaving the individual at a disadvantage when exposed to a new pathogen. We have directly tested this hypothesis using a mouse model of lifelong persistent infection.
There is also video of the presentation, which I think you'll find interesting - the research community is clearly within striking distance of a range of novel and effective repair methodologies for immune system aging: