There is no way to keep the popover window open, it will close whenever you navigate away as you have found.
If you are wanting to see the window purely to monitor the progress of the download perhaps to see when it has completed, you can create a saved search folder to do a similar thing which may help. Navigate to the download folder in Finder, and search for ",download", which is the extension given to files that are currently downloading. Save the search, call it "currently downloading" or something and add it to your side bar. Now, so long as you have this window open somewhere, you can view the progress of downloads, and if you use icon view the progress per download is displayed in the icon itself.
Of course, this does not retain a list of downloaded files after they finish which you need to clear every so often like the popover window, or provide download speed/time indications, but it may be a suitable workaround spending on your reasons for keeping it in view.