First off, what is "Adam's transgression"?
What that is referring to is when Adam and Eve were in the Garden of Eden and were told by God not to eat the fruit of the tree of knowledge. Long story short, Satan tempted Eve and she ate the fruit and then Adam did as well, or else he would remain in the garden alone.
Because Adam and Eve disobeyed Heavenly Father they were cast out of the Garden of Eden and could no longer be in the presence of God. They became subject to sin, pain, and death, which is why we are subject to those same things.
In the Church of Jesus Christ of Latter-Day Saints we do not believe we are held accountable for Adam's disobedience at the beginning of time.
It says in Deuteronomy 24:16 "The fathers shall not be put to death for the children, neither shall the children be put to death for the fathers: every man shall be put to death for his own
The only sins we believe we are accountable for at the last day are our own.
I know some people, including loved ones of mine, believe that we are under punishment because of Adam right from the get go. That's why some people believe their infants need to be baptized ASAP.
In the Book of Mormon one of the Prophets named Moroni went to the Lord concerning this matter and this was the answer he received:
8 "Listen to the words of Christ, your Redeemer, your Lord and your God. Behold, I came into the world not to call the righteous but sinners to repentance; the whole need no physician, but they that are sick; wherefore, little children are whole, for they are not capable of committing sin; wherefore the curse of Adam is taken from them in me, that it hath no power over them; and the law of circumcision is done away in me.
10 "Behold I say unto you that this thing shall ye teach—repentance and baptism unto those who are accountable and capable of committing sin; yea, teach parents that they must repent and be baptized, and humble themselves as their little children, and they shall all be saved with their little children.
We can see in this response from the Lord to Moroni that "children are alive in Christ" and that they aren't able to commit sin because they don't know right from wrong. Why would God punish an innocent child for something someone did 1000's of years ago? Would you punish one of your children for something that your Grandfather did?
Those children that have past away at the beginning years of their life are not lost. They are with our loving Savior Jesus Christ and our eternal Father.
We believe that we have a loving, unchangeable God who adores and cares so much for each individual that is, was, and will be on this earth. He wants all of us to return to live with Him through baptism at the age of accountability and will not punish us for something that someone else did.
25 Adam fell that men might be; and men are, that they might have joy. 2 Nephi 2:25