If you can access it, this paper offers a nice overview of the subject. The idea is essentially based off of the claim that:

Quote Originally Posted by Paul Crutzen
In the stratosphere, chemical and micro-physical
processes convertSO2 into sub-micrometer sulfate particles. This has been observed
in volcanic eruptions e.g., Mount Pinatubo in June, 1991, which injected some 10
Tg S, initially as SO2, into the tropical stratosphere (Wilson et al., 1993; Bluth et al.,
1992). In this case enhanced reflection of solar radiation to space by the particles
cooled the earth?s surface on average by 0.5 ◦C in the year following the eruption
(Lacis and Mishchenko, 1995). Although climate cooling by sulfate aerosols also
occurs in the troposphere (e.g., Ramaswamy et al., 2001), the great advantage of
placing reflective particles in the stratosphere is their long residence time of about
1?2 years, compared to a week in the troposphere. Thus, much less sulfur, only a
few percent, would be required in the stratosphere to achieve similar cooling as the
tropospheric sulfate aerosol (e.g., Dickinson, 1996; Schneider, 1996; NAS, 1992;
Stern, 2005). This would make it possible to reduce air pollution near the ground,
improve ecological conditions and reduce the concomitant climate warming. The
main issue with the albedo modification method is whether it is environmentally
safe, without significant side effects.
(emphasis mine)

For completeness' sake (or if the article is unavailable), I note that this paper is arguging that we might wish to add the sulfur to the upper atmosphere to compensate for the effect of our current efforts to reduce it from the lower atmosphere, an effect which will otherwise increase solar flux by ~0.10% per year, and is estimated would, alone, lead to a net 0.8K increase in global temperature. His estimate is that this would require about ~2% of the current global current SO2, and pegs the cost at a not-insubstantial $25-50 billion per year, although that claim is apparently based on 1992 technology.

The wikipedia article on stratospheric sulfur aerosols discusses the topic in marginal detail.