A little caution helps you stay safer online. For a public video, you should not need to give a downloader your Facebook password.
Check the link
Use the original Facebook video link and make sure it opens as expected. Be careful with unfamiliar pages that ask you to install software or enter private account details.
Use your own judgement
Download only content you have the right to use. Videos may include music, personal images or other material belonging to someone else.
Manage your files
Keep your phone or computer updated and remove files you do not need. If a video is unavailable or restricted, do not try to bypass its access controls.