I think they have a lot of influence, but I don't think the use of the phrase "these days" is correct.
Fans have been flocking to see their "idols" and passing out at their shows since The Beatles, Elvis and beyond.
Modern pop icons like Justin Bieber probably have bigger followings because the world is more connected than it used to be and media is instantly accessible across the globe.
I think that's why it may seem like celebrity influence has grown, but the truth is kids were probably dressing up like the Emperor of Rome back in 50 B.C. because they thought he was trendy.
We all look up to and try to imitate those who we perceive to be successful - it might be a survival instinct that's inherently part of us.
I guess the cause for concern these days is more "the message" rather than the reach.
I noticed Lady Gaga has recently had her septum pierced. And whilst I'm all in favour of piercings, I wonder how many of her "Little Monsters" (the name she's given to her adoring fans) will follow suit.
I guess, ultimately, if you're bringing up a child - it's your job to be the number 1 influencer in your kids life. It's natural that kids will be influenced by all sorts of positive and negative elements in their environment - but making sure they "turn out OK" is a parent's mission.