Bob,
That definitely sounds like you have some code in a media query that is hiding the pane. Check your CSS files in the skin for something that looks like this: @media screen and (max-width: 1000px) {}
Whatever the min/max width is set for should correspond to the browser width at which the pane disappears, so that could help you narrow it down if there are a lot of them. Depending on what the selector is for your left pane, there will be something like this:
.left-pane-class { display: none; }
Or maybe .left-pane-class { visibility: hidden; }
If there's nothing like that, it might be a more complicated error like if you have a bug in your grid layout and your wrapper has overflow: hidden; that could cause the pane to pop outside the wrapper and that would make it disappear. The only other thing I can think of is that JS could be running that detects the screen size and messes with the style of the pane or the removes it from the DOM.
I can't help you troubleshoot it very well without actually looking at the code, sorry.
Hope this helps.