Question: "How do we know that the Bible is the Word of God, and not the Apocrypha, the Qur’an, the Book of Mormon, etc.?"
Answer: The question of which (if any) religious text is the true word of God is of utmost importance. To avoid circular reasoning, the first question we must ask is: how would we know if God communicated in the first place? Well, God would have to communicate in a manner that people could understand, but that also means that people could make up their own messages and simply claim that they came from God. So, it seems reasonable to think that if God wanted to authenticate His communication He would have to verify it in a manner that could not be duplicated by mere humans - in other words, by miracles. This narrows the field considerably.
Beyond the evidence for the Bible's correctness (manuscript evidence) and its historicity (archeological evidence), the most important evidence is that of its inspiration. The real determination of the Bible's claim to absolute inspired truth is in its supernatural evidence, including prophecy. God used prophets to speak and write down His Word and God uses miracles like fulfilled prophecy to authenticate His messengers. For example, in Genesis 12:7, God promises that the land of Israel was to be for Abraham and his descendants. In 1948 Israel was returned back to the Jewish people for the second time in history. This may not seem so astonishing until you realize that no nation in the history of the world has been scattered from its homeland and returned! Israel has done it twice. The book of Daniel predicts with accuracy the coming of the four great kingdoms from Babylon, to Medo-Persia, to Greece, to Rome centuries before some of those kingdoms came on the scene (a time span of over 1,000 years!) with details concerning how they would rule and be broken. This includes the reigns of Alexander the Great and Antiochus Epiphanies.
In Ezekiel 26 we can see in astonishing detail how the city of Tyre was to be destroyed, how it would be torn down, and how its debris would be thrown into the sea. When Alexander the Great marched on that area, he encountered a group of people holed up in a tower on an island off the coast near there. He could not cross the sea, so he could not fight those in the tower. Rather than wait them out, the proud conqueror had his army throw stones into the sea to build a land bridge to the tower. It worked. His army crossed the sea and overthrew the occupants of the stronghold. But where did he get so much stone? The rocks that were used for the land bridge were the leftover rubble from the city of Tyre . . . its stones cast into the sea!
There are so many prophecies concerning Christ (over 270!) that it would take more than a few screens worth of space to list them all. Further, Jesus would have had no control over many of them such as His birthplace or time of birth. Second, the odds of one man accidentally fulfilling even 16 of these are 1 in 10^45. How many is that? For comparison, there are less than 10^82 atoms in the entire universe! And Jesus, who affirmed the Bible as the Word of God, proved His reliability and deity by His resurrection (an historical fact not easily ignored).
Now consider the Quran - its author, Muhammad, performed no miracles to back up his message (even when he was asked to by his followers - Sura 17:91-95; 29:47-51). Only in much later tradition (the Hadith) do any alleged miracles even show up and these are all quite fanciful (like Muhammad cutting the moon in half) and have zero reliable testimony to back them up. Further, the Quran makes clear historical errors. Muslims believe the Bible is inspired but with some errors from editing (Sura 2:136 as well as Suras 13, 16, 17, 20, 21, 23, 25). The question they cannot adequately answer is: "When was the Bible corrupted?" If they say before 600 A.D. then how can the Quran admonish believers to read it? If they claim it was after 600 A.D., then they have jumped out of the frying pan and into the fire, for we have absolutely no doubt as to the accuracy of biblical manuscripts from at least the 3rd century forward. Even if Christianity were false, the Quran still has an insurmountable problem because it makes judgments against Christians for believing things that they do not (nor have they ever) believed. For example, the Quran teaches that Christians believe the Trinity is the Father, the Mother (Mary), and the Son (Sura 5:73-75, 116), and the Quran also teaches that Christians believe that God had sex with Mary to have a son (Suras 2:116; 6:100-101; 10:68; 16:57; 19:35; 23:91; 37:149-151; 43:16-19). If the Quran is really from God, then it should at least be able to accurately report what Christians believe.
Joseph Smith, the author of the Book of Mormon, tried to do some miracles such as prophecy (a test for a true prophet in Deuteronomy 18:21-22) but failed several times. He foretold of Christ's second coming in History of the Church (HC) 2:382. Joseph Smith preached that the coming of the Lord would be in 56 years (about 1891). The second coming did not occur in 1891, and the Mormon Church does not claim that it did. Nor has it occurred since. He also prophesied that several cities would be destroyed in Doctrine and Covenants (D&C) 84:114-115. New York, Albany and Boston were to be destroyed if they rejected the gospel according to Smith. Joseph Smith himself went to New York, Albany, and Boston and preached there. These cities did not accept his gospel, yet they have not been destroyed. Another famous false prophecy of Joseph Smith was his "END OF ALL NATIONS" in D&C 87 concerning the rebellion of South Carolina in the war between the states. The South was supposed to call on Great Britain for aid, and as a result war would be poured out upon all nations; slaves would revolt; the inhabitants of the earth would mourn; famine, plague, earthquake, thunder, lightning, and a full end of all nations would result. The South finally did revolt in 1861, but the slaves did not rise up, war was not poured out upon all nations, there was no worldwide famine, plague, earthquake, etc., and there was no resulting "end of all nations."
The collection of writings that Protestants call the Apocrypha (hidden writings), Roman Catholics call the deuterocanonical (later or second canon) books. These books were written between 300 B.C. and 100 A.D., the Intertestamental Period between the inspired writings of God's Prophets in the Old Testament and those of the Apostles and their contemporaries in the New Testament. These were "infallibly" accepted into the Bible by the Roman Catholic Church in 1546 at the Council of Trent. Now the Apocrypha would be covered under the evidence for the Bible if these writings were truly inspired - but evidence seems to indicate that they are not. In the Bible we find prophets of God whose messages are ratified by miracles or prophecy that comes true, and whose message is immediately accepted by the people (Deut 31:26; Josh. 24:26; 1 Samuel 10:25; Daniel 9:2; Col. 4:16; 2 Peter 3:15-16). What we find in the apocrypha is just the opposite - no apocryphal book was written by a prophet. None of these books were included in the Hebrew Scriptures. There is no ratification of the authors of any apocryphal book. No apocryphal book is cited as authoritative by later Biblical writers. There is no fulfilled prophecy in any apocryphal book. Finally, Jesus, who quoted from every section of Old Testament Scripture, never once quoted from the apocrypha. Neither did any of His disciples.
The Bible so far outshines every competing source for being God's revelation that if it is not God's Word, it would seem impossible to choose among the leftovers. If the Bible is not God's Word, then we have been left with no clear criteria by which to know what might be.