This is a great question, and I will attempt to give you a satisfactory answer, but not before I ruin it with theory, deconstruction, and cultural analysis. A lot of prehistory, and especially the sort of lived experiences you are asking about, does not leave direct evidence of their occurrence for us to excavate and interpret. Archaeologically, infants leave little evidence beyond burials, and child rearing in many ways is a transient phenomenon. If it weren't for the bottles, breast pumps, diapers, toys, car seats, cribs, and the thousands of other products parents are compelled to buy, what physical evidence of you raising your son would there be in 20,000 years? Prehistorians rely on material culture (stuff people make) to have evidence of human actions in the past. Since prehistoric babies and parents don't leave much material culture of their interactions lying around, we have to resort to other tools to answer questions like these. Many prehistoric archaeologists are also anthropologists at heart, and use tools like ethnography (the study of extant human cultures), analogy, and theory to try and understand better what is the question we are asking and how can we address the question without direct evidence. This is what we must do here, and a few prehistorians have addressed your question along these same lines.

Your one month old cries a lot (so does my infant daughter, but it gets better!). Our concern is that if you lived in the Upper Paleolithic, around 25,000 years ago, a nasty pack of cave hyenas may hear him and decide that baby EpicBlueDrop would be an easy meal. It seems rather likely that people in the past were occasionally preyed upon by animals higher up the food chain. We have a variety of carnivore tooth marks found on bones throughout the Paleolithic of Europe, including cave bear, cave lions, and wolves, though it is often difficult to determine whether the bite marks were produced pre- or post-mortem. However, your fear is attested in a Chagga lullaby, “Do not cry, my child! What are you crying for? If you wail, the leopard will devour your mother.” [Here I had to pause because my daughter started to cry for a bottle. No leopards were seen.] But, it is clear that the crying of infants has been passed down evolutionarily to us today, so there must overall be better reasons for crying than there are dangers associated with it.

Eleanor Scott tells us that infants in "traditional" societies cry much less than those in western societies due to different ways of parenting. Infants generally cry for something they need, often food or comfort. [Another quick break here to go comfort my infant. She couldn't time these better.] Scott says that babies are continually carried or worn in slings by women much of the day and night, and suckled when needed. This almost continual comfort and feeding would minimize the need for an infant to cry. Scott also emphasizes the constancy of walking (a feature of Paleolithic and foraging life)with an infant on one's side, inducing a calming effect. Basically, babies cried less in these "traditional" societies because they were attended to more and less often left alone. We must also consider that Paleolithic babies would have had a community to support them, not just the parents. Levi-Strauss mentions a Tacuna myth in which the hero Monmaneki and his wife went off to hunt, leaving their infant in the care of its grandmother, to sleep in her arms. So even when the mother must be away from her child, someone else in the community manages the care. Collective care (and even collective nursing), would have been typical for many of your Paleolithic ancestors, rather than relying on simply mom and dad. Scott claims that the relationship with which human infants were accustomed to throughout prehistory, in which their needs were met without frequent crying and a near constant companionship, has been fractured in western cultures. I believe this gets at two parts of your question: your baby (and mine) cries more because of the way our culture practices parenting, and people in the past were taking care of their babies just fine, if not "better" in the sense that they cried less than ours.

But we still have the lingering problem of why infants and babies cry, and cry so loudly and piercingly, given the off chance that leopards/hyenas might find you. This gets into questions of evolutionary biology, for which I am less prepared. Luckily, again Scott has thought about this, suggesting that shrieking cries are a signal that someone needs to come attend to the needs of the infant, now. Infants that have urgent needs met, be it prevention of starvation or leopard issues, tend to survive better than infants who do not have their needs met. If we accept that in our "traditional" Paleolithic society that infants cry less because their needs are well attended, then a wailing baby may signal something it seriously not right. In response, mom and the other adults in the group would spring to action. Given the survival of the infant, the cries were successful in ensuring its survival, and would be a useful trait to continue to appear in future populations.

Ok, so then the answer to your question. If your baby lived in prehistoric times (assuming Paleolithic), wouldn't he have been gobbled up by now by an opportunistic cave critter? Maybe, but. Your baby probably cries more than your typical Paleolithic baby, because you and I parent differently than our ancestors. Also, when a Paleolithic baby screamed, there were several people very nearby to do something about it, whether that was providing food or fighting a hyena leopard. On a final, less happy note, prehistoric babies died all the time. Like I mentioned at the beginning, burials are the most archaeologically visible aspect of infancy and childhood we have, even in the Paleolithic, even with Neanderthals. It is difficult to judge the rate and causes of mortality for infants and children, but death was an all too common feature for young children in much of the past. While predation was likely a cause of death for infants, we have little evidence for actual causes of death in prehistoric infants.

If you are interested, Eleanor Scott's book The Archaeology of Infancy and Infant Death was the source for much of the theoretical answer here, specifically related to crying. Güner Coşkunsu's volume The Archaeology of Childhood: Interdisciplinary Perspectives on an Archaeological Enigma has a broad discussion of other topics of childhood in archaeology, including the visibility of children, children's roles in prehistoric society, childhood death and burial in prehistory, and even evidence of kids playing in the Ice Age.

While we have users here who do focus on prehistory, you may want to x-post your question to our sister-sub /r/AskAnthropology, as there is a good deal of overlap on the topic with their focus there.