Did the Nephilim that are mentioned in Genesis 6:4 do anything beyond just existing?
The Nephilim are a subject of much debate. There are a number of views on what they were, the two prominent views being the Sethite view and the Angelic view. Both are an opinion on who or what the "sons of God" are.
The Nephelim are most notably mentioned in Genesis and seem to be a driving decision to send the Flood of Noah.
A few things stand out in this one verse:
Generally, the Sethite view holds that the Sons of God were the righteous men descended from Seth; they praised and worshiped God. The daughters of men are women descended from Cain, who came from wickedness. This view contends that there is nothing particularly special about these persons and it is meant to highlight that the mixing of the righteous people of Seth with the wicked people of Cain led to further wickedness and their subsequent total destruction by the Flood.
Generally, the Angelic view holds that the Sons of God were angels or some other spiritual creatures that procreated with human women. This view contends that this act alone was wicked and an indication that these angelic beings were fallen from God's grace, therefore the offspring were inherently wicked. This view also contends that the offspring were giants and beyond human capabilities.
With both of these views, there is very little Biblical support. However, I find the suppositions and theories very interesting and even fun to research in a science fiction kind of way.
The extent to which your exact question is answered depends on which view you want to hear. From the Sethite view it pretty much stops at Goliath, if you call him a Nephilim because he was a giant. From the Angelic view, we can go all the way through Revelation. Outside of the Bible we can make many suppositions and links to mythology, but that is a bit outside of the scope of this site.
Here are some of the major points concerning the Nephilim:
Points from the scripture
Points from the Sethite View
Points from the Angelic View
Wikipedia has a fantastic article on the Nephilim, however, it seems very disorganized to me.
*I wrote an essay on the topic, but I could not find it. I did find my preliminary notes. Here they are.
The following are all the verses that are or might be related to the Nephilim.
Gen. 6:4—The Nephilim were on the earth in those days—and also afterward—when the sons of God went to the daughters of men and had children by them. They were the heroes of old, men of renown.
The following were preliminary conclusions. They are from a biblical literalist view.
The Nephilim were a real people who lived before and after the flood. That is very clear in the Scripture. They received the name Anakites from Anak who was a descendant to Arba who was "the greatest man among them." The land they lived in was called Kiriath Arba, but it was later called Hebron. The mention of the Emites leads me to conclude that these people were also from the Nephilim, but perhaps their lineage was lost or unknown. Dt. 2:10 shows the confusion of who they were, but it strongly asserts that they were also large and strong people. "Rephaites" is only mentioned in passing in a few areas. It is possible, that because of confusion and different people using different names, that Anakites, Emites, and Rephaites were all the same people; it is also possible that Gittites were the same people as well. Dt. 9:2 hints that there were other tribes of people who may have been related to the Nephilim through the Anakites, or they may be directly related to the Nephilim. Perhaps like Israel, they were divided into tribes, maybe from the sons of Arba or Anak. The mention of Ahiman, Sheshai and Talmai in Nu. 13:28 and Jos. 15:13,14 hints at this possibility; it is not a stretch to assume that these three persons were chiefs, kings, or some other kind of leaders fr their people. When we look at Goliath the scriptures clearly call him and his family Rephaites, who are descendants of Anakites. There are quite a few details about them, and they are all surrounded by stories of war with the Philistines. The Philistines clearly were allied with them or perhaps the Rephaites were Philistines. Considering the Philistines were eventually completely destroyed, the last of the descendants of the Nephilim were either killed or integrated into other, normal races. If there were (or are) surviving Nephilim the Bible does not explicitly record it.
The Nephilim were on the earth in those days, and also after that, when the sons of God came in unto the daughters of men, and they bore children to them; the same were the mighty men that were of old, the men of renown. - Genesis 6: 4
Over time, Genesis 6: 4 generated a mass of speculative oral traditions. The reasons are fanciful translations of the word 'Nephilim', and the terms 'sons of God' and 'daughters of men', and innovative interpretations of the verse. The correct translation for 'Nephilim' is 'giants' but there is a play on words as 'nephyl' means 'fallen', so giants that have 'fallen' or 'come down'. Using a bit of imagination the 'fallen' was turned into 'fallen angels', aliens in every form, and even the 'survivors of abortions'. 'Sons of God' can as easily be translated as 'sons of the gods', meaning, followers of the false gods or 'sons of royalty' with the explanation that the children of the rulers were the Nephilim. It was even taken to mean the descendents of Seth while 'daughters of men' has the obvious meaning of could mean 'the descendents of Cain'.
The wide variety of translations made it possible for a number of interpretations to emerge and some of them have even generated a new genre of fiction. One of the lesser known but still interesting explanations for the verse involves Neanderthal man.
Neanderthal man was very similar to humans except that they were much stronger with huge muscles and had great endurance. The other big difference was that he had a larger brain and the manner of his burial rites show spirituality, and care of the old and sickly shows social conscience. Neanderthal managed to survive in conditions so cold that even an Eskimo would freeze without knowing how to sew by making coats and duvets of feathers. (Tying feathers together to make a coat was well within their capabilities and from there the belief in angels with wings.) To primitive Homo sapiens, he must have seemed like a giant.
The Carmel Caves on Mount Carmel are amongst the most important of the prehistoric caves in Israel as they were continuously occupied for an incredible 150 000 years. Here prehistoric man went from cave dweller to a builder of homes. From a hunter-gatherer to a planter and domesticator. And most importantly here the remains of mixed humans were found - half Neanderthal, half Homo sapiens. (Some scholars called this ‘human’ Palaeanthropus Palestinensis (Ancient Palestinian Man)) Here was the beginning of ancient man using ‘modern’ (Mousterian) Neanderthal tools and adopting a 'modern' culture that contained the first religion.
For 90 000 years the two groups lived side by side without much interaction but eventually Neanderthal came down from the mountains and mixed with the humans resulting in the mighty men that were of old, a new people that had an advanced culture. From the Neanderthal, man learnt how to work metals and arrange flowers, how to hunt as a team, and how to make war, how to cure the sick and take care of the old, and most significantly man adopted their religion that centered on funeral rites. Eat the flesh; bury the bones. The bones for eternal life in the hereafter, and the flesh so the departed stays part of the survivors, and so enjoy eternal life on earth. To the humans this ritual must have looked like cannibalism and that explains Enoch 7: 4 (The giants turned against them and devoured mankind.)
From this idea, a more elaborate hypothesis evolved that has some sound science mixed with some very strange thinking. Humankind throughout its development split up (the Tower of Babel is an example) and individual groups developed for periods in isolation. From time to time, these groups would get together again and merge to form a new group. Genesis 6 describes such coming together of three groups and there is enough archeological evidence to make it believable that this happened in Israel. The three groups were:
• The 'sons of God' are those who served, or were supposed to serve, God. They were the descendents of Seth. The use of 'sons' is to show those that toil on the earth from generation to generation without much progress. From Seth came hope for the future, living for future generations, and belief in the unseen.
The new group that formed from the mixing of these three is what is called 'modern man', Homo sapiens sapiens, man aware and creative. There is a confirmation of 'man creative' in verse 5, if 'imagination' is taken to mean 'creative thinking'. (And the Lord saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually.) 'Imagination' is translated from the Hebrew 'yetzer' and there is 'yetzer hara' or creativity/imagination that leads to evil and 'yetzer tob', creativity/imagination that leads to good. In Genesis 8: 21, God said in His heart that no longer will humanity as a whole be punished for the sin of Adam (that is what made man aware) and neither will there be a group punishment for the 'yetzer hara' that had now become part of man.
Madeleine L'Engle gives a possible opinion in her novel "Many Waters": https://en.wikipedia.org/wiki/Many_Waters. She shows them interacting with the humans during the time of Noah.
Adding this at the prompting of David Stratton, though I think the Wikipedia entry will be more useful and factual for readers: as I recall (it's been a number of years since I read it), L'Engle portrays nephilim as shadowy, quasi-evil individuals of great beauty, power, and persuasion. There are examples of intermarriage with humans of the time and something about it being difficult to deliver the child of a nephil. When I read the book, I had not yet come across nephilim in any other context and remember wondering if she was portraying fallen angels.