For instance, i have arachnophobia (fear of spiders). Would they have this? Or would they be used to traveling through spider webs, having spiders in their homes, perhaps even eating spiders? What about water? Some people fear water they can not see the bottom of, or just drowning. But would these people be used to much more dangerous things and not care about a little water?
My question boils down to this: Have people become cowards?
It's certainly true that some conditions of life can become normalized to the point people no longer talk about them. Even moving from one geographic area to another can be a case of this. A great historical example is itchiness. Lice, fleas, mosquitoes, were endemic in the pre- and early modern (European) world. We know from discussions and material evidence that people were indeed bothered by them and tried to get rid of the infestations, but even those serve as a vital reminder of the underlying life conditions of everything else that we talk about.
As far as fears go, though: I'm not qualified to talk about anthropological or psychological studies of "primal fears." You can definitely trace a deep-rooted fear of the ocean in premodern Western sources, however. Alain Corbin's The Lure of the Sea is a wonderful (if limited by his use of literary sources) exploration of the changing relationship of humans and ocean. But it's not an accident that when late antique and medieval writers need a metaphor for the world gone sour, their go-to language is "the shipwreck of the world." Medieval Christian sailors carried consecrated Eucharist hosts and saints' relics with them to try to ward off storms and sea monsters. And as /u/Georgy_K_Zhukov and I recently discussed, the story of Jonah and an anxiety to prove that humans could survive bad events in the ocean preoccupied 19th and early 20th century Americans and Europeans.
That's just one example to get you started! The study of fear as a concept and of individual fears throughout history has an incredibly rich body of literature. If you're interested in reading about the sort of premodern fear that is largely (not entirely) absent from the modern world, have a look at this answer on death and hell in the late medieval imagination: