What Happens to Babies Who Die?

Sinclair Ferguson on a difficult doctrine: Total Depravity in Children

Elisha Raises the Shunammite woman's Son: Frederic Leighton

The total depravity of our children is a faith-doctrine, a biblical insight. Our natural instinct is to think of new-born children as moral and spiritual tabulae rasae, clean sheets on which to write a successful life. Admittedly the page may soon be a little blotted (the occasional temper tantrum!), but the background is still basically white, surely? Not so, according to the Scriptures: the wicked go astray from the womb and speak lies from birth, insists the psalmist.

Here is a chart comparing Arminianism and Calvinism. I'll excerpt the part about our sin-nature.
Arminianism Free Will or Human Ability says:

Although human nature was seriously affected by the fall, man has not been left in a state of total spiritual helplessness. God graciously enables every sinner to repent and believe, but He does not interfere with man’s freedom. Each sinner possesses a free will, and his eternal destiny depends on how he uses it. Man’s freedom consists of his ability to choose good over evil in spiritual matters; his will is not enslaved to his sinful nature. The sinner has the power to either cooperate with God’s Spirit and be regenerated or resist God’s grace and perish. The lost sinner needs the Spirit’s assistance, but he does not have to be regenerated by the Spirit before he can believe, for faith is man’s act and precedes the new birth. Faith is the sinner’s gift to God; it is man’s contribution to salvation.

Calvinism: Total Inability or Total Depravity says:

Because of the fall, man is unable of himself to savingly believe the gospel. The sinner is dead, blind, and deaf to the things of God; his heart is deceitful and desperately corrupt. His will is not free, it is in bondage to his evil nature, therefore, he will not — indeed he cannot — choose good over evil in the spiritual realm. Consequently, it takes much more than the Spirit’s assistance to bring a sinner to Christ — it takes regeneration by which the Spirit makes the sinner alive and gives him a new nature. Faith is not something man contributes to salvation but is itself a part of God’s gift of salvation— it is God’s gift to the sinner, not the sinner’s gift to God.

Jan Sluijters (1881-1957), Elisha and the son of the
Shunammite woman, 1904

Total depravity does not mean that all humans are all as bad as they could be. Not everyone is a Hitler, is he? There are people who are nice and compassionate and do kind things. Then there's the Hitlers and Stalins and Pol Pots of the world. No, total depravity means that sin has touched every part of our being, totally, and we are totally unable to do anything good for God. We are all criminals in God's eyes and in His capacity as Judge He would be righteous if He sent all humans to hell upon their death. Even children.
So what happens to children who die, if they are completely depraved but are unable to consciously and maturely respond to the Gospel?
One of the complaints I receive about the Doctrines of Grace is that God would be horribly mean to send babies to hell when they do not have the mind to even be able to contemplate the Gospel and formulate a response to it, the basis of faith and the key to entry into the Kingdom. I agree, that would be hard to swallow. But God does not send babies or children to hell. He is just, and He is sovereign, and He has a plan.
After the terrorist attacks on September 11, 2001, John MacArthur was invited to the Larry King talk/interview show to try and help people make sense of the spiritual questions the event raised. In a subsequent sermon titled What Happens to Babies Who Die?, Dr MacArthur said this:

Some of you who tuned in to the Larry King Show a week ago Saturday will remember that Larry fired a question to me on the air that came out of nowhere. The question that reveals a nagging, troubling issue in the human heart. He asked me, "What about a two-year-old baby crushed at the bottom of the World Trade Center?"
I answered, "Instant heaven."
He replied with another question. "Wasn't a sinner?"
I again answered, "Instant heaven."
That's a compelling question, what about a baby crushed at the bottom of the Trade Center? What about any baby that dies? It's an agonizing question. It's a question that plagues Christians and non-Christians alike...what happens to babies that die?

In the essay, MacArthur shows from scripture that they go to heaven. Though they are sinners from the womb and though they have not hard or responded to the Gospel, God made a provision.

The Doctrines of Grace make me love Jesus all the more. He made a provision for the children and adults otherwise unable to cognitively understand their sin or the Gospel. God is love and His provision is demonstrates no better than the very moment an apple cheeked youngster's laugh is stilled by death, when He receives them instantly to His arms.
Of Children:
The seed of every known sin is planted in their hearts. Robert Murray McCheyne
Of Jesus:
Never forget that there is more grace in Christ than there is sin in your heart and your child’s heart combined. Sinclair Ferguson
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Further reading:
These essays show from scripture why their authors believe children who die go to heaven.
Charles Spurgeon: Infant Salvation
R. Albert Mohler, Jr. and Daniel L. Akin: The Salvation of the ‘Little Ones': Do Infants who Die Go to Heaven?
John Piper: What Happens to Infants Who Die?