For some kinds of files, such as executables (.exe), Firefox will not show the Open option. But for documents (.doc, .pdf, .xls) and archives (.zip, .rar), Firefox should show the Open option.

If you never get the option, it's possible that the file which stores download associations has become corrupted. The standard solution is to rename the file and have Firefox regenerate it from scratch with the default settings. You will need to recustomize some choices, for example, if you do not want to use the built-in PDF viewer.

Start by opening your current Firefox settings (AKA Firefox profile) folder using either

• "3-bar" menu button > "?" button > Troubleshooting Information
• (menu bar) Help > Troubleshooting Information
• type or paste about:support in the address bar and press Enter

In the first table on the page, click the "Show Folder" button. This should launch a list of files and folders in a new window.

Leaving that window open, switch back to Firefox and Exit, either:

• "3-bar" menu button > "power" button
• (menu bar) File > Exit

Pause while Firefox finishes its cleanup, then rename mimeTypes.rdf to something like mimeTypes.old

Start Firefox back up again. Any improvement in the download dialog?

And/or, in case one of your extensions is involved, could you test downloading in Firefox's Safe Mode? That's a standard diagnostic tool to deactivate extensions and some advanced features of Firefox. More info: Diagnose Firefox issues using Troubleshoot Mode.

You can restart Firefox in Safe Mode using either:

• "3-bar" menu button > "?" button > Restart with Add-ons Disabled
• (menu bar) Help > Restart with Add-ons Disabled

Not all add-ons are disabled: Flash and other plugins still run

After Firefox shuts down, a small dialog should appear. Click "Start in Safe Mode" (not Reset).

Any difference?

My mistake. I thought I had tested it on files other than executables. Evidently not.

Wound up installing the OpenDialog2 extension, which works around it nicely. Thanks for your help, everyone.