Is there a way to add a wrapper class to my blocks?
If I understand you correctly, you can use the Theme Developer module to inspect the block itself.  You will be able to take the default block template out of the main theme directory, rename it accordingly, and then edit it to have your class.  This is not the best way to achieve this result, but it will work. Make sure to disable the module when you are done using it. 

In addition to that, the block themselves will have their own independent ids "#block-block-17".  A little more painful to work with, but if you don't have that many it could be another solution for you.