Workaround instructions from Google:

Usually mentioned issue appears because of missing Adobe Flash Player installed. You have to know that Adobe Flash Player is directly integrated with Google Chrome and enabled by default and that's why you don't experience issues with it. You can find more information on how the Flash Player works with Chrome at the following Help Center article:

As a fix I will recommend you to install Flash Player by following the steps below:
1. Visit
2. Select "Do you have a different operating system or browser?"
3. Once you press the option a new window will appear saying
"Install a different version of Adobe Flash Player". You will see drop-down menus and you will be able to set it.
Once you do, restart your browser and try again to attach file.

In case the instructions here don't work, please open a new Firefox Window and in the location bar type:
1. About:config and press Enter.
2. Select "I'll be careful, I promise".
3. Check for network.http.spdy option and if you find such one 
4. Double click on in order to disable.
This will disable the SPGY protocol which causes issues when attaching files in Gmail.

The same steps are valid for your Opera browser. But instead about:config in the location bar, you have to type opera:config.