You may be old-fashioned in the eyes of some, but that doesn't make you wrong! You should be concerned about what your children watch, and I'm afraid much of what goes on in the media today is morally and spiritually corrupt.
Never forget: Our children learn how to live not only by what we say but also by what we do (or fail to do). We may tell them certain things they see or hear are wrong, but if we fail to back up our words with our actions, then why should they believe us? We wouldn't let our children eat garbage or drink poison because we're concerned about their physical welfare. Why shouldn't we be just as concerned about their spiritual welfare — which is greatly influenced by what their minds and hearts consume? The Bible warns, “I am afraid that just as Eve was deceived by the serpent's cunning, your minds may somehow be led astray” (2 Corinthians 11:3).
Instead, encourage your children to enjoy wholesome entertainment and activities. You can't shield them from everything that's bad in the world, nor should you; if you did, they might not recognize it and turn from it when they're older. But don't just put restrictions on them; explain why you're doing it.
Most of all, ask God to help you be an example to your children of what it means to follow Christ. God's will is that we might “become blameless and pure, children of God without fault in a crooked and depraved generation” (Philippians 2:15).