I am somewhat surprised by the findings, but there is a lot of published research on this topic.
A paper published in Science in 1976 maintains that:
There is convincing evidence that both negative and positive ions (i) inhibit growth of bacteria and fungi on solid media; (ii) exert a lethal effect on vegetative forms of bacteria suspended in water when opportunity is provided for contact of cells and ions; and (iii) reduce the viable count of bacterial aerosols
Moreover, the same paper concludes that "in the case of mice infected with influenza virus, ion-deprivation increases the cumulative mortality rate" and "since ion depletion is a constant concomitant of modern urban life, one reasonably may speculate about comparable inimical effects on humans."
A study published in the Journal of Applied Microbiology in 2009 finds ionization "to be an effective method for reduction in surface and airborne bacteria." In this study ionizers were installed inside refrigerators, and although effective the amount of reduction depended "the constructional layout and material properties of the ionizer housing, air circulation within the refrigerator and the interior volume" or the refrigerator.
An earlier study in the Journal of Food Protection concludes that "high levels of negative air ions can have a significant impact on the airborne microbial load, and that most of this effect is through direct killing of the organisms."
By contrast, a study done in 2007 by the School of Civil Engineering, University of Leeds suggests that "the bactericidal action attributed to negative air ions by previous researchers may have been overestimated."
To summarize then, it seems that ionizers can indeed reduce the airborne bacterial load but could also be bad for humans. A reasonable piece of advice comes from the same 1976 article in Science quoted above, which recommends to "maintain air ion concentrations and ratios approximating those existing in nature" and expresses hope that "air ion levels of urban air will approach those of clean rural air."