There's a couple of ways you can tell if you've shifted focus from the column pane to the data grid pane. When you are working in any of these panes, JMP will shift 'focus' to that pane. So if you click on the column pane, select your columns, etc, then it is in focus. You'll see the gray bar just over the column pane (middle left pane) darker than the lighter gray bars over the data table pane (top left), row pane (bottom left), and data grid itself (everything to the right of those 3 panes). Keep hitting tab and you'll see the dark gray shift from pane to pane to pane. From the column pane if you hit tab 2x it will bring you first to row pane and then to data pane. You can then copy your data (or data + column names) with ctrl+C or ctrl+shift+C.
The other way to tell is to look at the column names (assuming any of the selected columns are in view, which may not be the case for wide tables). When the pane is highlighted they are a deep blue (or whatever color). when the pane is not selected, they are a lighter shade. The first method above is a more reliable way to know what is in focus.