Were they watching the saga of the Ring and its outcome with keen interest or were they disinterested and washed their hands of the whole affair? After the way West was bent and effectively closed and the dominion of man decreed for rule of Middle-earth, did they 'really' care what happened on Middle-earth?
Just to be clear - I'm not sure if you know this, based on your question - the term Eldar applies specifically to those Elves and their descendants who received and followed the summons of the Valar to Valinor.
By the time of the War of the Ring, very few Eldar remain in Middle Earth. These Eldar are certainly highly aware and interested in the War of the Ring. But the Eldar no longer exist in large numbers, and so we wouldn't expect them to project military force. The only known remaining settlements of the Eldar are:
- Lothlorien. There can be no doubt that all the Eldar in Lothlorien were gravely concerned with the War of the Ring:
‘Some there are among us who sing that the Shadow will draw back, and peace shall come again. Yet I do not believe that the world about us will ever again be as it was of old, or the light of the Sun as it was aforetime. For the Elves, I fear, it will prove at best a truce, in which they may pass to the Sea unhindered and leave the Middle-earth for ever. Alas for Lothlórien that I love! (Haldir, The Lord of the Rings)
For if you fail, then we are laid bare to the Enemy. Yet if you succeed, then our power is diminished, and Lothlórien will fade, and the tides of Time will sweep it away. We must depart into the West, or dwindle to a rustic folk of dell and cave, slowly to forget and to be forgotten.’ (Galadriel, Lord of the Rings)
In addition to the skirmishes discussed in the book with Orcs, Lothlorien was also at actual war during the War of the Ring:
Three times Lórien had been assailed from Dol Guldur, but besides the valour of the elven people of that land, the power that dwelt there was too great for any to overcome, unless Sauron had come there himself. Though grievous harm was done to the fair woods on the borders, the assaults were driven back; and when the Shadow passed, Celeborn came forth and led the host of Lórien over Anduin in many boats. They took Dol Guldur, and Galadriel threw down its walls and laid bare its pits, and the forest was cleansed. (Lord of the Rings, Appendix B)
- Mirkwood. You can also have little doubt that the Eldar in Mirkwood were intensely concerned with the War of the Ring and the rest of Middle Earth, since the king's son was a member of the fellowship. Mirkwood was also involved a low-level war with Dol Goldur, which culiminated with a battle at the end of the Lord of the Rings:
In the North also there had been war and evil. The realm of Thranduil was invaded, and there was long battle under the trees and great ruin of fire; but in the end Thranduil had the victory. (Lord of the Rings, Appendix B)
Rivendell. Again, it's hard to see how the Eldar here could be disinterested: Elrond is a key figure in the War of the Ring, and Elrond's sons fought at Pellennor, the Eldar were on the Council, and Glorfindel saved Frodo from the Ringwraiths. The Eldar here also played a role in getting the Witch-King of Angmar out of the North.
The Havens. We know least about the Eldar of the Havens, but we can imagine there are probably few permanently living there, and those that are concern themselves mostly with building boats for the Elves leaving. Nevertheless, we do know Cirdan, the lord of the Havens, held one of the Three Rings and gave it to Gandalf.
‘Take this ring, Master,’ he said, ‘for your labours will be heavy; but it will support you in the weariness that you have taken upon yourself. For this is the Ring of Fire, and with it you may rekindle hearts in a world that grows chill. But as for me, my heart is with the Sea, and I will dwell by the grey shores until the last ship sails. I will await you.’ (Lord of the Rings, Appendix B)
The last group of Eldar we know the least about, because they did not return from the Undying Lands to tell us what they thought, with one exception - Glorfindel, who came with special dispensation from the Valar, and prior to the removal of Aman from the Cricles of the World.
Did these Eldar care? Absolutely. Especially the Noldor, who blamed themselves for the whole thing, but all the Elves loved Middle-Earth (maybe not so much the Vanyar, who seem to have not cared for many thousands of years), and the Teleri and the Noldor all had family there. The very forging of the Rings was an act of sin (or at least error) borne from a too-great love for Middle-Earth.
In the first we see a sort of second fall or at least ‘error’ of the Elves. There was nothing wrong essentially in their lingering against counsel, still sadly with3 the mortal lands of their old heroic deeds. But they wanted to have their cake without eating it. They wanted the peace and bliss and perfect memory of ‘The West’, and yet to remain on the ordinary earth where their prestige as the highest people, above wild Elves, dwarves, and Men, was greater than at the bottom of the hierarchy of Valinor. They thus became obsessed with ‘fading’, the mode in which the changes of time (the law of the world under the sun) was perceived by them. They became sad, and their art (shall we say) antiquarian, and their efforts all really a kind of embalming - even though they also retained the old motive of their kind, the adornment of earth, and the healing of its hurts. (Letters)
But there was nothing they could do. After the Changing of the World, Elves would not get from the Undying Lands to Middle Earth, and at any rate, they weren't allowed to.