Filter with checks for "like" and "dislike". "Like" options sort closer to front, "Dislike" options don't appear at all.
Take existing categories, give each one a setting: "like", "indifferent", "dislike". When considering videos, each video tag with a "like" setting gives it a higher rank, and it comes out closer to the front. Each video with any "dislike" tag is immediately rejected and ignored. "Indifferent" tags don't work into the sorting algorithm.
You could extend this to a range of "how much I like it", to break ties, but the simple Yay/Boo system should be easier to implement and for people to learn.
And, as suggested elsewhere, a user's choices on those settings should be remembered in their user profile.
For the main page, where new videos are shown sorted by date, the "dislike" settings should still be used to filter out stuff that user just doesn't care about.