To the untrained reader, and especially to someone whose first experience of it is in a debate with an atheist, Psalm 137:9 is a very troubling verse. It's certainly not something you'd expect to find in "the good book", and lacking context, it can really cause problems. But this is actually one of many verses that teaches us the importance of taking verses in context.
I chose the header image for this article very carefully. The more well versed you are in memes, the more likely you are to recognise it. Usually, an image is presented that lacks normal intelligence. A screenshot of a dumb Twitter post, for example. The man shown in this picture is then displayed underneath with the caption "kill yourself".
Ignoring the moral implications of such infantile memes, my point is that those who make or post them likely do not genuinely want the person they're mocking to commit suicide. It's a disgusting insult, nothing more. Now, if this is how internet trolls treat silly people on the internet, imagine how a Psalmist might speak about an invading army of terrorists.
This is exactly what's going on in this short Psalm. The Psalmist, exiled from Jerusalem, is lamenting the fact that he has been captured by such barbaric brutes. The Babylonians, about whom the Psalm is written, have done exactly the sorts of things he is now wishing on them, including dashing their infants against the rocks. Thus, whether the Psalmist meant it or not, what he is actually writing about is payback. What Babylon did to Israel, the Psalmist wants done right back to Babylon.
Vengeance is a very human desire, and at least on occasion, it reaches the height of wishing violence not against the perpetrator themselves, but against their loved ones. "The heart, Osborne! First we attack his heart!" You see, then, how much sense it makes for the Psalmist to express such a desire. And of course, when talking to God, it is ok to express one's heart. After all, it's not as if He doesn't know it anyway.
It is worth noting that God actually takes a different approach. Because He shows no partiality (Deuteronomy 10:17), and takes no pleasure in the death of the wicked (Ezekiel 18:32; 33:11), He is actually reluctant when He does enact vengeance. As we read in Lamentations 3:33: "For He does not afflict willingly, Nor grieve the children of men." Thus, we can be sure that the only way God would ever actually grant such a request is if it played a greater roll in the grand scheme of things.
Of course, this is where we tend to recoil and let emotion take over. The thought of doing something so horrible to a baby is just horrific to us - unless we do it 73 million times a year and call it healthcare... But yes, as painful as it is, God will do things we would consider terrible in the short term in order to ensure greater things in the future.
Of course, we do have three extra things to consider, all of which distinguish God from men. The first is that God is omniscient. He knows every possible ending of every possible choice He makes. Let's take the common time travel trope of going back in time and killing Hitler. This would probably be every Jew's dream, but if they were able to realise it, then everyone in 1889, not knowing the monster Hitler would grow up to be, would consider the time traveler a monster for killing an innocent baby boy. So, God knows what will happen if He does vs. doesn't kill any particular person at any particular point.
The second, following from this, is that God literally has the power and authority to grant every child that dies eternal life, which the Bible all but explicitly teaches He does. In other words, the infants of judged cultures are better off now than they would have been had God allowed them to grow into rebels like their parents.
The third is that God has the authority to do literally anything. He owns it all. Every life He takes, He gave in the first place. And not just as a mere vessel, like we do; He literally created everything from nothing. Thus, it is His life to take. This makes it even more amazing that His default is life in the first place. Knowing what He knew, He could have decided not to create the world, not to let Adam and Eve life for half a second after blinking in the direction of the forbidden fruit, not to let humanity stay fertile... His default is life. So much so that, once again, He takes no pleasure in the death of the wicked.
In fact, He actually lay down His own life. As scary as it is that God would even allow verses like Psalm 137:9 get into Scripture (much less inspire it), how amazing is it that God would endure such pain for His enemies! As human beings, we have earned our death sentence. To sin against the Author of life is as foolish as to sin against one's own heart. Cut your throat, is it your throat's fault you die?
But Jesus died to bring those who deserve death eternal life. There is no one on Earth so sinful that God will not forgive their sins. That is, if they repent. Nothing but faith can save us. Our options are confess Jesus as Lord, and believe He rose from the dead, or suffer a fate far worse that Babylon.