There is not one verse that says "Babies go to Heaven" or anything like that, so a look at the entire scope of the Bible is needed to answer the question. Earlier this year (before my wife got pregnant) I became more convinced than ever, that when a baby dies it goes to Heaven. I came to this conclusion from many implicit references in the Bible (again there are no explicit references about this).
I decided to give you the arguments and writings of one who has already written about this (no need re-inventing the wheel, right?). So I recommend the following resources to you from Pastor John MacArthur.
You can listen for free to his sermon series, "What Happens to Babies Who Die?" (which include Jesus Loves The Little Children, and two sermons that get deeper into answering the question that I really really recommend listening to both The Salvation of Babies Who Die, Part 1 & The Salvation of Babies Who Die, Part 2 )
If you are more of the book nerd type then you can get this in book form (called, "Safe in the Arms of God" which I am reading right now). The book starts off:
The question was fired at me by Larry King. I had been invited to participate as a panel member on the Larry King Live television program one Saturday evening. The program was taped in the aftermath of the September 11, 2001, attacks on the United States. Even though we had been discussing issues of life and death, grief and hope, as part of this program, Larry's question seemed to come out of nowhere.
"Instant heaven," I immediately replied.
Larry fired back a second question: "He wasn't a sinner?"
I again answered, "Instant heaven."
Larry's compelling questions revealed a nagging, troubling issue in the human heart.
What is the future of a baby crushed by the rubble of the collapsing World Trade Center? What about any baby that dies? What happens to an unborn child, an infant, a child, or even a physically mature but mentally handicapped adult with the mental capacity of a child after he dies? What is the fate of that "little one" as he or she enters eternity? The questions are agonizing ones for many parents, Christians and non-Christians alike.
A number of rather strange and ill-founded answers have been given to these questions in the past. The correct answer, however, begins very simply: "Instant heaven."
In the sound-bite environment of a program such as Larry King Live, I had no opportunity to follow up with an explanation for my statement and, frankly, Larry didn't ask for one. He seemed satisfied with my quick and decisive response and went on to other questions related to the ways in which our nation was grieving and recovering from the aftermath of that terrible day of tragedy.
But I believe you deserve an expanded answer because in all likelihood, you are reading this book after having experienced the loss of a child-or perhaps because you are a person in a position, as I have been too many times as pastor, to counsel or encourage someone who has lost a child. My heart grieves with any parent who loses a child, and that is what motivated me to search Scriptures on this subject so I could reach out and offer biblically founded words of comfort and encouragement...
After examining the Biblical evidence, Vanessa and I have been convinced that when a child dies they go to Heaven. This is consistant with the orthodox Christian view that we are saved by grace but damned by works. If you disagree, don't have a clue, or are not sure why you hold that position I would really recommend the above resources.
Soli Deo Gloria!