I have heard this claim often in discussions and also read it several times on this site:
The Bible cannot be properly understood without God's help.
Is there a scriptural basis for this claim?
Jesus said in Matthew 11:25, "I thank you, Father, Lord of heaven and earth, that you have hidden these things from the wise and understanding and revealed them to little children." By these words we know that God has knowledge to share that we can't get anywhere but from Him. This is reinforced by Jesus saying in Matthew 11:27, "All things have been handed over to me by my Father, and no one knows the Son except the Father, and no one knows the Father except the Son and anyone to whom the Son chooses to reveal him."
In addition, Paul wrote in 1 Corinthians 2:9-13: that we cannot comprehend the things of God unless His Spirit reveal them to us, and that is exactly what He has done to those who love Him:
But, as it is written, “What no eye has seen, nor ear heard, nor the heart of man imagined, what God has prepared for those who love him”—
these things God has revealed to us through the Spirit. For the Spirit searches everything, even the depths of God. For who knows a person’s thoughts except the spirit of that person, which is in him? So also no one comprehends the thoughts of God except the Spirit of God. Now we have received not the spirit of the world, but the Spirit who is from God, that we might understand the things freely given us by God. And we impart this in words not taught by human wisdom but taught by the Spirit, interpreting spiritual truths to those who are spiritual.
In Galatians 1:12, Paul tells us how he came by the gospel: "For I did not receive it from any man, nor was I taught it, but I received it through a revelation of Jesus Christ." This is an amazing admission by a man who was a master of his religion and of its holy book. He who sought to kill the disciples of Christ asserts that God had to reveal information not available to his natural mind in order to grasp its truths.
These are some of the arguments that we need God in order to apprehend spiritual truths, such as those in the Bible.
Some would use this phrase to say: "The Bible cannot be properly understood without God's direct intervention." There is no scriptural basis for this. Consider king Josiah, who found the scriptures in a dusty vault after years of disuse. Upon reading them, he understood them and took immediate action (2 Kings 22). It's possible to read scripture and understand it correctly without divine confirmation or signs.
But I think the phrase is more generally used to dissuade someone from thinking they understand scripture simply because they've read it and thought about it--that God plays a part in your understanding. In this sense, yes, there is absolutely scriptural basis.
First, God is within his power and rights to give understanding and take it away. 2 Corinthians 4:4-6 depicts that God has blinded some to the truth, but illuminated it to others. See also John 12:39-40 and Romans 9:14-18. One cannot simply read scripture and expect to understand immediately because God can choose to reveal or not. (I believe God values our choices when choosing whether to reveal truth, but that's a much longer discussion).
Second (and perhaps more pragmatically), understanding in the scripture always involved taking action. Understanding without action is no understanding at all. James 1:22-25 and 2:14-17 make this very clear. And this is why, in John 8:31-33, despite the Jews having believed him, Jesus challenges them to "hold to my teaching" because "then you will know the truth". We cannot properly understand (or believe) scripture without also applying it to our lives. If we apply it to our lives, we are choosing to accept God's help and so the statement is true.
I think it more clear to say "The Bible cannot be properly understood without trying to understand it God's way."
Yes, there is.
St. Paul talks about this distinction between the words written in the Word of God, which could be understood any number of ways, linguistically (even so as to seem foolishness), and the spiritual, that is, the inspired sense intended by God and conveyed in them (discerned and knowable only about the spiritually enlightened: enlightened by the Spirit who revealed the Word, Who alone knows the mind/intentions of God).
He distinguishes the mere rational mind (the wisdom of the world, devoid of the spiritual) and the spiritually enlightened mind (true wisdom; the wisdom of God):
1 Corinthians 2:6-14 (DRB) [parentheticals mine]
Howbeit we speak wisdom among the perfect: yet not the wisdom of this world, neither of the princes of this world that come to nought; 7 But we speak the wisdom of God in a mystery, a wisdom which is hidden, which God ordained before the world, unto our glory: 8 Which none of the princes of this world knew; for if they had known it, they would never have crucified the Lord of glory. 9 But, as it is written: That eye hath not seen, nor ear heard, neither hath it entered into the heart of man, what things God hath prepared for them that love him. 10 But to us God hath revealed them, by this Spirit. For the Spirit searcheth all things, yea, the deep things of God. 11 For what man knoweth the things of a man, but the spirit of a man that is in him? So the things also that are of God no man knoweth, but the Spirit of God. 12 Now we have received not the spirit of this world, but the Spirit that is of God; that we may know the things that are given us from God. 13 Which things also we speak, not in the learned words of human wisdom; but in the doctrine of the Spirit, comparing spiritual things with spiritual. 14 But the sensual [worldly] man perceiveth not these things that are of the Spirit of God; for it is foolishness to him, and he cannot understand, because it is [something which can only be] spiritually examined.
Notice that St. Paul teaches that the Holy Spirit alone can understand the mind of God. This excludes men without the help of that Spirit Who alone can understand God's revelation—and help the Church do so (John 16:13; 1 Timothy 3:15). Sometimes this means using His instruments. in the form of the Pastors of the Church (Acts 8:30-31; Ephesians 4:11).