“If I just touch his clothes, I will be healed.” The healing of the bleeding woman is depicted in the Catacombs of Marcellinus and Peter in Rome. (Mark 5, Matthew 9, Luke 8)
We help ourselves by following God, doing good works in faith, seeking righteousness and His will.
It's the most-quoted Bible verse that isn't actually in the Bible! My theory is that it likely originated from typically American pragmatism: the outlook that “what works is right” and (more generally); the can-do, self-reliant, nothing-is-impossible, rugged individualist American ethos. Pragmatism is no test of truth, but (like most philosophies or indeed, virtually any worldview whatever) it has truthful elements in it.
As I pondered this saying when I heard it trotted out again recently, while on vacation, I decided to explore whether anything in the Bible can be said to approximate the sentiment. I think a case can be made that several aspects of biblical teaching approach the content of this famous maxim. One fun way to do that is to do some word searches, with the help of the marvelous technology available today, to quickly search Holy Writ.
It was said of King David (1 Chr 12:18: RSV, as throughout) that “your God helps you,” but it’s not explained why, in the immediate context. We know, however, that David was a “man after God's own heart.” This is stated by St. Paul, as recorded in Acts 13:22: “. . . he raised up David to be their king; of whom he testified and said, `I have found in David the son of Jesse a man after my heart, who will do all my will.'” Paul was paraphrasing 1 Samuel 13:14: “. . . the LORD has sought out a man after his own heart; and the LORD has appointed him to be prince over his people . . .” Thus, by virtue of such cross-referencing, one might argue that God helped David because David followed God with all his heart.
Isaiah the prophet states that God helps him (Is 50:7, 9). I suppose that an obedient prophet has done quite a bit to be in a place where God would readily help him. Following God (like David also did) is, after all, manifestly helping oneself. 2 Chronicles 18:31 informs us that “Jehosh'aphat cried out, and the LORD helped him.” Jehosh'aphat helped himself by crying out to God, and God responded by helping. 2 Chronicles 26:7 tells us that “God helped” King Uzziah “against the Philistines” and other enemies. Why? Two verses earlier, we find the answer: “He set himself to seek God . . . and as long as he sought the LORD, God made him prosper.” King Uzziah sought God, which was also helping himself, because in so doing, it comes about that God helps the person who seeks Him.
It all comes down to what it means to help oneself. Biblically speaking, seeking God is always doing that. More directly to the point is King David's Psalm 37:39-40:
The salvation of the righteous is from the LORD;
he is their refuge in the time of trouble.  The LORD helps them and delivers them; he delivers them from the wicked, and saves them, because they take refuge in him.
The motif of divine “rewards” offers more relevant biblical data:
1 Samuel 26:23 The LORD rewards every man for his righteousness and his faithfulness; . . .
2 Samuel 22:21 The LORD rewarded me according to my righteousness; according to the cleanness of my hands he recompensed me. (Ps 18:20 is exactly the same in RSV)
Hebrews 11:6 And without faith it is impossible to please him. For whoever would draw near to God must believe that he exists and that he rewards those who seek him.
Likewise, the same is the case with God’s “blessing”:
Proverbs 3:33 The LORD's curse is on the house of the wicked, but he blesses the abode of the righteous.
I think we see a general pattern here: God blesses or helps or rewards those who, in faith, seek Him and seek to be righteous. The Bible and the Catholic faith also teach that human beings can do no good thing without being enabled by God's grace beforehand. So it's a scenario of men being enabled by God to be good and follow Him, and then subsequently deciding with their own free will whether to continue doing so or to cease. It's not either/or. It's synergistic: God working with man / man working with God. This brings us to the glorious biblical theme of being God's “co-laborers”":
Mark 16:20 And they went forth and preached everywhere, while the Lord worked with them and confirmed the message by the signs that attended it. Amen.
1 Corinthians 3:9 For we are God’s fellow workers; you are God’s field, God’s building.
1 Corinthians 15:10 But by the grace of God I am what I am, and his grace toward me was not in vain. On the contrary, I worked harder than any of them, though it was not I, but the grace of God which is with me.
2 Corinthians 6:1 Working together with him, then, we entreat you not to accept the grace of God in vain.
Philippians 2:12b-13 . . . work out your own salvation with fear and trembling; for God is at work in you, both to will and to work for his good pleasure.
Thus, we see that the essential meaning of “The Lord helps those who help themselves” is taught in the Bible, and, moreover, that it's emphatically not works-salvation (Pelagianism). Rather, we help ourselves by following God, doing good works in faith, seeking righteousness and His will. He Himself brings about these desires in us, by His grace, and enables us to carry out such good resolves, by His grace and the power of God the Holy Spirit: the Helper, or Counselor (Jn 14:16, 26; 15:26; 16:7; Rom 8:26; 1 Jn 2:1).
Because of our good works, by His grace (Heb 13:1-5), He helps us all the more: “Hence we can confidently say, 'The Lord is my helper,’ . . .” (Heb 13:6). The famous saying is not a Bible passage, but it does contain much biblical wisdom and truth.