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.
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.