DETAILS:
http://www.jobkeeperalliance.org/
WordPress 3.6.1
Atahualpa 3.7.12

This particular site has an entire background image. I would like to have the background of my sidebar be white with 40% opacity, but when I do the following under ATO > Style & Configure Sidebars > RIGHT sidebar style:

border-left: 2px solid #c7cbce;
padding: 20px 0 20px 0;
margin:10px 0 10px 0;
background:#FFFFFF;
opacity:0.4;
filter:alpha(opacity=40); /* For IE8 and earlier */

It makes everything in the sidebar transparent. Seeing that, I tried adding the following under ATO > Style Widgets > Widget Container

opacity:1 !important;
filter:alpha(opacity=100); /* For IE8 and earlier */

This did nothing.

Do I just need to poke around in the code to find the right selector or am I missing something else? Any help is always greatly appreciated! Thanks!

As I understand it, opacity is cumulative, you are making a cell of a table (the sidebar) opaque and everytinhg in that cell is affected.
you could try a semi transparent png as the sidebar background

I literally just said "Duh." to myself out loud after reading the latter part of your reply, Juggle.

I should have thought of the transparent png. Thanks!

You can also use

background-color: rgba(0, 0, 0, .5)

This would give you 60% opacity of a black background.