So I'm watching the LOTR trilogy and I'm on Fellowship of the Ring. It seems as if the Nazgûl actively avoid water, such as at the Bucklebury Ferry and the river at Rivendell. Does water harm the Nazgûl or lessen their powers in any way? Or are they just fussy and don't want to get their natty black robes wet?
In The Unfinished Tales, Christopher Tolkien wrote a passage concerning their fear of water.
All except the Witch-king were apt to stray when alone by daylight; and all, again save the Witch-king, feared water and were unwilling, except in dire need, to enter it or to cross streams unless dryshod by a bridge.³
³ At the Ford of Bruinen only the Witch-king and two others, with the lure of the Ring straight before them, had dared to enter the river; the others were driven into it by Glorfindel and Aragorn. [Author's note.]
And later on:
My father nowhere explained the Ringwraiths' fear of water. It is made a chief motive in Sauron's assault on Osgiliath, and it reappears in detailed notes on the movements of the Black Riders in the Shire: thus of the Rider [...] seen on the far side of Bucklebury Ferry just after the Hobbits had crossed [...] it is said that "he was well aware that the ring had crossed the river; but the river was a barrier to his sense of its movement, and that the Nazgûl would not touch the 'Elvish' waters of Baranduin. [...]" My father did indeed note that the idea was difficult to sustain.
This confirms that they avoided water out of fear, that their fear was not related to their mounts, but was deeper than that. It is theorized widely that the purity of the water in Middle-earth was an affront to their nature, but this is unconfirmed.
The Silmarillion states that all water is under the control of Ulmo, the second most powerful of the Valar:
Ulmo is the Lord of Waters. He is alone. He dwells nowhere long, but moves as he will in all the deep waters about the Earth or under the Earth. He is next in might to Manwë, and before Valinor was made he was closest to him in friendship; [...] For all seas, lakes, rivers, fountains and springs are in his government; so that the Elves say that the spirit of Ulmo runs in all the veins of the world.
Also, Ulmo never "abandoned" the Eldar and Men, even after the fall of Númenor:
Nonetheless Ulmo loves both Elves and Men, and never abandoned them, not even when they lay under the wrath of the Valar.
The Nazgûl, being wraiths, may fear water, but they are not harmed by water. However, their mounts can be drowned. That is why the wraith avoids attempting the jump onto the ferry and opts for the long way around. The Black Rider didn't fear for himself. He feared that he might lose his horse, knowing he could not replace it until another could be obtained from the stables of the Dark Lord himself.
Later, when the Nazgûl are overwhelmed in the river on the way to Rivendell, we learn that the Nazgûl are unharmed by the flood, but their horses are drowned and that after their horses were lost, the nine Wraiths were forced to return to Mordor on foot (which gained Frodo and company some much needed time) where they replaced them with bigger, badder, flying mounts.
This quote from Book 2, ch.1 explains why the loss of a Nazgûl mount was problematic:
"Because they are real horses; just as the black robes are real robes that they wear to give shape to their nothingness when they have dealings with the living."
"Then why do these black horses endure such riders? All other animals are terrified when they draw near, even the elf-horse of Glorfindel..."
"Because these horses are born and bred to the service of the Dark Lord in Mordor. Not all his servants and chattels are wraiths!"
Bear I mind that, traditionally, water is the realm of naiads, who are potentially related to the good elves. While this mythology is not explored in depth in LotR, the basis of it would have been well known to JRRT, and so may have been the justification for water – running water especially – being a force on the side of good. The ford of Bruinen looks very much like a working out of this type of mythology.
And, of course, it is a mythology. The fact that there are two occurrences where the dark riders seem to be wary of water – which may be for very different reasons, although as @Gabe indicates, they are both elvish waters – this is enough to establish a myth that they are afraid of the water. Personally, I would not want to risk my life on that.