This is a hereditary condition, with no treatment. One would have to process human blood to get IgA to inject into people, but the half-life of IgA is only five days. Given that it is not a life-threatening proposition (while getting blood products can be!), this idea hasn't caught on.
IgA is related to gamma globulin (IgG), which is what they shoot you up with if you are exposed to hepatitis (or rabies). There is a whole family of immunoglobulins that all help fight infections in different places; IgG is the most important one.
Fortunately for me, I have lots of IgG. This means that while I might be sick and miserable, I'm probably not going to die.
I have heard that people with IgA deficiencies are less prone to allergies: that allergies are an over-reaction to nasties, and colds are an under-reaction. I do know that I catch colds just by looking at a sick person, and am not allergic to even one thing. My mother has really bad allergies, and never catches colds.
I have decided that I am optimally evolved for small farming communities. Unfortunately, that is not where I hang out.
People who are IgA deficient are more prone to cysts, for some odd reason, and this may account for the cyst in my jaw that ultimately led to jaw surgery, but that is a different story.