Moroni 7:16 For behold, the Spirit of Christ is given to every man …

16 For behold, the Spirit of Christ is given to every man, that he may know good from evil; wherefore, I show unto you the way to judge; for every thing which inviteth to do good, and to persuade to believe in Christ, is sent forth by the power and gift of Christ; wherefore ye may know with a perfect knowledge it is of God.

17 But whatsoever thing persuadeth men to do evil, and believe not in Christ, and deny him, and serve not God, then ye may know with a perfect knowledge it is of the devil …

This passage comes from the prophet Mormon’s great dissertation on faith, hope, and charity, which comprises the entirety of Moroni chapter 7. Moroni, in completing the compilation of sacred records, was inspired to draw from his father’s writings during a period of Mormon’s ministry prior to the great final dissolution of the Nephite political entity, during which he was able to preach to and nurture what remained of the congregation of the true believers in Christ. He begins his comments with a discussion of the manner by which people may judge between good and evil, quoting Jesus, who taught the principle that “by their fruits ye shall know them” (Matt 7:16,20; 3 Ne 14:16,20)  which follows from the fact that neither can goodness flow out of a corrupt source, nor evil from a good and godly one. But buried in the middle of this exposition is the exceptionally bold assertion that every person has been given the power to discern good from evil, which is called “the Spirit of Christ.” So all have, or have had this capacity. Twice, in this sermon, Mormon includes the warning to be careful about how one judges between good evil. Apparently, it is not always easy, for he cautions in verses 14 and 18:

 14 Wherefore, take heed, my beloved brethren, that ye do not judge that which is evil to be of God, or that which is good and of God to be of the devil.

18 And now, my brethren, seeing that ye know the light by which ye may judge, which light is the light of Christ, see that ye do not judge wrongfully; for with that same judgment which ye judge ye shall also be judged.

Since Christ is he who has bestowed this gift of discernment upon mankind, it should come as no surprise that it is also Christ who shall be the final judge of how each person has used or misused it (Rom 2:16; 2 Tim 4:1; 2 Ne 9:41; Morm 3:20). 
In the Book of Mormon, every doctrine discussed is recapitulated  in one way or another by at least once, as far as I have been able to account for. This doctrine is no exception, except that the exact phrasing is only used in one other passage, this time by Moroni, who, in the previous passage was quoting from his father Mormon's sermon, now uses this phrase somewhat differently in his discussion of spiritual gifts in Mor 10:17, where he states:

17 And all these gifts come by the Spirit of Christ; and they come unto every man severally, according as he will.

18 And I would exhort you, my beloved brethren, that ye remember that every good gift cometh of Christ.

The idea that every person is given the ability to discern good from evil is consistent with the statements of other Book of Mormon writers, as well as New Testament writers. Consider these cross-references:

2 Ne 2:26-27

26 And the Messiah cometh in the fulness of time, that he may redeem the children of men from the fall. And because that they are redeemed from the fall they have become free forever, ...

 27 Wherefore, men are free according to the flesh; and all things are given them which are expedient unto man. And they are free to choose liberty and eternal life, through the great Mediator of all men, ...

In verse 26 the cause-effect assertion is that the redemption wrought by the Messiah is the source of men's ability to have knowledge of good and evil. Verse 27 recapitulates that idea in terms of the freedom to choose through the great "Mediator of all men" who is the Messiah.

Hel 14:30-31

30. ... for behold, ye are free; ye are permitted to act for yourselves; for behold, God hath given unto you a knowledge and he hath made you free.  

31 He hath given unto you that ye might know good from evil, ...

Here the source of the knowledge of good and evil is referred to more generically as God, who, as we learn from other passages, generally refers to Christ, who is the God of Israel (3 Ne 11:14) and God over all the earth (1 Ne 11:6, 13:41).
This direct assertion that all men are enlightened by the Spirit of Christ, Light of Christ, gift of God, etc. is consistent with this very direct statement in the Gospel of John:

John 1:9,17

9. [Jesus] was the true Light, which lighteth every man that cometh into the world.

17. For the law was given by Moses, but grace and truth came by Jesus Christ.

John plainly asserts that Jesus Christ enlightens every soul and that all goodness and truth flow from him.

Modern revelation makes the same assertion in even plainer language.

D&C 84:45-46; 93:2

45  For the word of the Lord is truth, and whatsoever is truth is light, and whatsoever is light is Spirit, even the Spirit of Jesus Christ.
46  And the Spirit giveth light to every man that cometh into the world; and the Spirit enlighteneth every man through the world, that hearkeneth to the voice of the Spirit.

2 And that I am the true light that lighteth every man that cometh into the world;

The idea that all men are ultimately accountable to Jesus Christ because all goodness and truth, and the ability to recognize them come through and from him and the power of his atoning sacrifice, is one of those doctrines that I consider one of the big central religious ideas which the Book of Mormon has greatly clarified.