Unfortunately there is no theme option for that so you will need to make a modification to the theme. Preferably with a child theme, or if you are using Jetpack, it has a nice css editor. But the theme styling for this is in the style.css file around line 1281 and looks like this:
[code type=css]
header.comment-meta {
margin-bottom: 14px;
background-color: #f3f4f4;
font-style: italic;
}
So when you make an override for header.comment-meta, you just need to copy and paste the above but just have the background colour like this:
[code type=css]
header.comment-meta {
background-color: #f3f4f4;
}