Thanks! It turns out that the div ID changes every time the page loads and is different for every listing (details) page.
I wrapped the description in the div as you suggested:
<div class="hide-my-custom-div">
Content to be hidden
</div>
I was then able to hide description for all grid views by targeting the
class and then the newly created one:
/*Hiding description in Nearby Places section of Details page*/
.geodir-gridview .hide-my-custom-div {
display: none;
}