Would adding sulfur dioxide to the atmosphere have a global cooling effect?
There have been a range of studies on the issue published in Atmospheric Chemistry and Physics and other scientific journals.

The short answer is "yes." The longer answer is "yes, but...."

In Climatic impacts of stratospheric geoengineering with sulfate, black carbon and titania injection, Jones, et. al. (2016) note: 


  In this paper, we examine the potential climatic effects of
  geoengineering by sulfate, black carbon and titania injection against
  a baseline RCP8.5 scenario. We use the HadGEM2-CCS model to simulate
  scenarios in which the top-of-the-atmosphere radiative imbalance due
  to rising greenhouse gas concentrations is offset by sufficient
  aerosol injection throughout the 2020–2100 period. We find that the
  global-mean temperature is effectively maintained at historical levels
  for the entirety of the period for all three aerosol-injection
  scenarios, though there is a wide range of side-effects which are
  discussed in detail.


Those side-effects would vary with the size of the particles used and the method and location of atmospheric injection, but would include:


Impacts on human health from increased particulates.
Changes in surface albedo from particulate deposition. 
Changes in the amount and location of precipitation across the globe. 
Increased stratospheric warming.
Changes in stratospheric ozone levels.


There would be no amelioration for other climate impacts such as ocean acidification.


  It is therefore important to note that the safest possible solution to
  avoiding the sort of climate change instantiated by, e.g., Fig. 6a of
  this report is to effectively mitigate greenhouse-gas emissions.


Other issues not addressed by this physics-based paper include:


The requirement for any geoengineering scheme to be maintained for decades or centuries.
The need for a substantial industrial base to manufacture and distribute the particulates involved.
The (presumed) need for an international body to manage the process and address political concerns such as the costs and benefits of the   geoengineering; such as if the process was going to increase or reduce rainfall to a particular region.