YouTube's terms of service (section 5) unless their is a download link provided by YouTube (present for some public domain and Creative Commons licensed videos). Making a screencast of a YouTube video is also a violation of the terms of service as well as being a likely copyright infringement.
So if there is a video that you see on YouTube that you think you or your students would like to reuse in another project, get in touch with the person who uploaded it and ask for permission to use a copy. Otherwise, take a look at the following three sources of public domain video clips.
The Internet Archive is the first place that comes to mind when I am asked for a source of Public Domain media. The Moving Image Archive within the Internet Archive is an index of more than 1.7 million video clips. Most of what you will find in the Moving Image Archive can be downloaded in a variety of file formats. You can search the archive by keyword or browse through the many categories and thematic collections in the archive. One important thing to note about the Internet Archive is that you probably don't want students to search it without supervision. In fact, I'd probably just create a folder of footage from archive that I share with my students.
Flickr is known for hosting images, but it also hosts video clips. Use the advanced search functions in Flickr to find video clips that have been released into the public domain and to find videos that have a Creative Commons license attached to them.
The Public Domain Review is a website that features collections of images, books, essays, audio recordings, and films that are in the public domain. Choose any of the collections to search for materials according to date, style, genre, and rights. Directions for downloading and saving media is included along with each collection of media.