Yes, a lot can go wrong in life, and the Bible certainly doesn't tell us to close our eyes to its dangers. The Bible says, "A prudent man sees danger and takes refuge, but the simple keep going and suffer for it" (Proverbs 22:3).
But the Bible also warns us against becoming "worry warts" (to use your expression), or being filled with worry and anxiety over everything that might go wrong. I suspect that if you made a list of all the things you worried about in the last year, almost none of them actually happened. And yet your worries robbed you of joy, and probably made your family miserable as well!
Instead, the Bible tells us to trust God for the future. You see, when we allow worries to overwhelm us, we're actually saying to ourselves that God must not love us, and He must not be watching over us. But this isn't true! God loves you far more than you can ever imagine–and the proof is that He was willing to send His Son into the world to die for your sins. He loves you so much that He wants you to spend eternity with Him in heaven.
The Bible says, "Do not be anxious about anything, but in everything, by prayer and petition, with thanksgiving, present your requests to God. And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus" (Philippians 4:6-7). I challenge you to put these words into action–for when you do, God's peace will begin to overcome your worries.