The word might is the most important operative word in this famous saying, not the resultant word joy. Adam fell that men might be. Adam and Eve did not know all things—they did not know exactly how they could fulfil this commandment to multiply and replenish the earth once they were cast out. It took a great deal of faith for Adam to choose to fall.
Until there was a fall, we were just waiting. There was no way for the great family of God to receive bodies until after the fall of Adam. Because Adam fell, we are and by his fall came death. By reason of transgression cometh the fall and the fall bringeth death. The fall was to bring mortality, and men were placed on earth to bring to pass the immortality and eternal life of man. The great purposes of God would have been frustrated without the fall of Adam.
And then Lehi says, "men are that they might have joy." Lehi knows from his vision of the tree of life that a joyous outcome is not guaranteed but must be chosen. Initially, Adam says more definitely in Moses 5:10: "for because of my transgression my eyes are opened, and in this life I shall have joy." Adam is confident that he shall or will have joy. And in verse 11, in her response, Eve states more specifically that the joy to be experienced was, at least initially, "the joy of our redemption": his wife Eve answered, "Were it not for our transgression we never should have had seed [children], and never should have known good and evil, and the joy of our redemption, and the eternal life which God giveth unto all the obedient."
Before the Fall, they were not mortal; after the fall they were mortal. Before the fall they were not subject to physical death; after the fall they were subject to physical death. That is one of the things Satan lied about, "Ye shall not surely die" (Genesis 3:4). Before the fall they were in the presence of God; after the fall they were cast out or cut off from the presence of God. Before the fall they were spiritually alive; after the Fall they suffered a spiritual death—they died spiritually, they died as pertaining to things of righteousness, they were cut off from the presence of God. That is what spiritual death is. Before the fall they were in a state of innocence; after the fall they had knowledge. Before the fall they did not know the difference between good and evil; after the fall they knew the difference between good and evil. Before the fall they would have had no children; after the fall they had children. So let us vote. How many people believe that the fall was good? Yes! And yet so many Christian believers do not have that understanding. The fall was a great blessing. It was part of the plan—first the creation in a certain condition, then the fall changed that condition. We said they are subject to death and they are subject to spiritual death. Now we need to do something to save them and to redeem them, and that is why Lehi immediately follows with the message of a Savior.
Book of Mormon Central, "Why Did Lehi Teach that the Fall was Necessary? (2 Nephi 2:22–25)," KnoWhy 269 (February 1, 2017).
Book of Mormon Central, "What are the Origins of Lehi’s Understanding of the Fall? (2 Nephi 2:25)," KnoWhy 28 (February 8, 2016).