The speaker wants the picnic to stay dry. Choose the standard sentence with hope.
I don't hope it rains during the picnic.
I hope it doesn't rain during the picnic.✓Correct answer
I hope not it rains during the picnic.
I doesn't hope it rains during the picnic.
Explanation
With hope, the desired negative outcome is expressed inside the following clause. Moving the negation onto hope would normally say that the speaker lacks hope, which is a different meaning.