I've been hearing YouTube referred to as the new Napster. Had it gone on longer, it may have gotten itself in more trouble with broadcasting copyrighted stuff.
That said, without Broadband - I barely watch it. I see a few here and there on my computer in my office if someone has linked to one that looks interesting, I'll watch it. But I don't spend time searching it. I rely on word of mouth and even then, I don't often have the time to watch.
Edit:
Google's Copyright Nightmare - this contradicts what I wrote in the first sentence. It might be worse.