You're asking how 'Christians can base their belief on something that's written in a book'. Simple answer: they don't 'base' their belief on what's written in a book. Their belief is based on the workings of two things: 1) the Holy Spirit, and 2) Faith. It is those two things that give us certainty and a peace. I began my life as an atheist, but eventually became curious about the way the world worked and why we did things the way we did. Jesus says, 'Behold, I stand at the door and knock; if anyone hears My voice and opens the door, I will come in to him and will dine with him, and he with Me'. Once you sincerely seek God, his Holy Spirit which inhabits this world today enters you and reveals the nature and reality of God to you.
Simply put, the living God is there for every man and woman, but you have to be open hearted in seeking him. Once you do that, he creates lifelong assurance and certainty of his presence, power and eternal love for you. If you are not willing to ask God to reveal himself, openly and with humility, then no amount of 'searching' for another answer in life will succeed. Instead, you will go through life being blind to the very power that created and loved you. I cannot think of a sadder reality than that.
The Bible is God's word revealed to man. His spirit and my faith are the power that reveal himself to me. If you do not possess either, it is as though you are blind and will not see. That is the key difference between belief and being lost.