We must get off the mountain! Make for
the gap of Rohan and take the West road
to my city.
The Gap of Rohan takes us too close to
We cannot pass over the mountain. Let us
go under it. Let us go through the mines
of Moria. Gandalf has a concerned look on
Let the ringbearer decide.**
From the film Lord of the Rings: Fellowship of the Ring (script) http://www.imsdb.com/scripts/Lord-of-the-Rings-Fellowship-of-the-Ring,-The.html
This scene is not inside the books.
A likely but speculated answer would be:
Peter Jackson added it to emphasise on the importance of Frodo as the ringbearer. Essentially, wherever Frodo goes, the Company would follow him, as he is the one destined to carry the ring and deliver it to Mordor. Elrond defines the roles of the other 8 here:
The others go with him as free companions, to help him on his way.
Lord of the Rings: Fellowship of the Ring - The Ring goes South
This is supported in a QnA session:
Question: When the Fellowship are on the side of the mountain and arguing about which direction to take, Gandalf says "Let the ring bearer choose". Why does Gandalf say that? He knows that Frodo has enough on his plate, what with taking the ring all the way to Mordor, so why add to his problems by making Frodo decide the way to go?
Answer: Because, like it or not, he's the leader of this quest. Gandalf is only a guide; Aragorn,Boromir,Legolas, and Gimli are Frodo's guardians, while the other three Hobbits are simply companions. Besides, Frodo's the one carrying the increasingly burdensome ring. If he thinks one path easier than another then that's his call.
In the book however, it is a different matter.
On the pass of Caradhras, the Company is assailed by the bad weather: biting winds and lots of snow.
he heard Bilbo's voice speaking. I don't think much of your diary, he said. Snowstorms on January the twelfth: there was no need to come back and report that!
Later on in the day (night-time then) when dawn came, they made the decision.
Gimli looked up and shook his head. 'Carahras has not forgiven us,' he said. 'He has more snow yet to fling at us, if we go on. The sooner we go back and down the better.'
To this all agreed, but their retreat was now difficult. It might well prove impossible.
In the book, the decision to retreat back down Caradhras was made by the whole Company, and not solely by Frodo or Gandalf.