(11/26/13 update: This post has been revamped to include three, not two, methods of minimizing explicit results. Also, the title has been changed to reflect the slightly different focus of the post.)
Here are three ways to minimize your chances of accidentally pulling up explicit content when showing a YouTube video to your students in class.
1. First of all, choose a video ahead of time, and bookmark that link. Test the link just before you show the video.
Doing a spontaneous YouTube search in class is risky. This is especially true with viral videos that seem to attract less-than-honorable people who want to reroute you to their explicit websites.
2. Secondly, if you want to give the video a more permanent home, you can embed it on your website.
Make sure the "Show suggested videos when the video finishes" option is unchecked.
3. Thirdly, which is the most drastic but surest method, you can enable "Safe Search" in Google to filter all explicit text and images.
Here's a little demo I recorded of how to add and lock "Safe Search" settings.
Google notes that even though it tries to eliminate explicit content in Safe Search, "We do our best to keep SafeSearch as up-to-date and comprehensive as possible, but objectionable content sometimes slips through the cracks."
You can check out Google's Safe Search support page for information on how to report explicit content.