Yes, I believe they do. The only reason that God damns some people is that because they are sinful. Jesus Christ came to save us from that inevitable sinfulness. Babies, on the other hand, cannot sin. They don’t have the knowledge of the law.
“What shall we say then? Is the law sin? God forbid. Nay, I had not known sin, but by the law: for I had not known lust, except the law had said, Thou shalt not covet. But sin, taking occasion by the commandment, wrought in me all manner of concupiscence. For without the law sin was dead. For I was alive without the law once: but when the commandment came, sin revived, and I died. And the commandment, which was ordained to life, I found to be unto death. For sin, taking occasion by the commandment, deceived me, and by it slew me. Wherefore the law is holy, and the commandment holy, and just, and good. Was then that which is good made death unto me? God forbid. But sin, that it might appear sin, working death in me by that which is good; that sin by the commandment might become exceeding sinful” (Romans 7:7-13).
In this way, babies cannot sin. God will not damn the one who did not even know sin, nor even have an opportunity to sin! Therefore, babies are saved.