There was nothing particular about the elves that made them capable of taking on balrogs, except that the elves, as the Firstborn children of Illúvatar, were probably the most powerful of all the races native to Arda (meaning, not counting Ainur like Gandalf). What was more important was that the greatest warriors—the kind who might have faced down a balrog—were, by and large, no longer to be found in Middle-earth.
Gandalf is explicit that no one else in the Company, including the elf lord Legolas, is a match for the balrog.
‘Fly! This is a foe beyond any of you. I must hold the narrow way. Fly!’
Indeed, Legolas, who normally seems to have ice water in his veins, is quite un-manned (or un-elfed) by the sudden appearance of Durin's Bane.
Legolas turned and set an arrow to the string, though it was a long shot for his small bow. He drew, but his hand fell, and the arrow slipped to the ground. He gave a cry of dismay and fear. Two great trolls appeared; they bore great slabs of stone, and flung them down to serve as gangways over the fire. But it was not the trolls that had filled the Elf with terror. The ranks of the ores had opened, and they crowded away, as if they themselves were afraid. Something was coming up behind them. What it was could not be seen: it was like a great shadow, in the middle of which was a dark form, of man-shape maybe, yet greater; and a power and terror seemed to be in it and to go before it.
‘Ai! ai!’ wailed Legolas. ‘A Balrog! A Balrog is come!’
Tolkien's Middle-earth is fundamentally a fallen world, in which the greatness of the past is always ebbing away, because of the way the world had been corrupted by Morgoth. Everything is less than it once was, and this manifests itself in many ways. For example, the greatest creations of the elves and Ainur, about which most of Tolkien's stories ultimately revolve—whether the Two Trees, the Silmarils that contained their light, or the Rings of Power—are artifacts that can only be made once; the power to duplicate them no longer exists.*
Similarly, there were many more heroes who might have challenged a balrog during the Elder Days. Ecthelion and Glorfindel each killed a balrog during the Fall of Gondolin, although they lost their own lives in the process.** Yet they were just two among the twelve greatest nobles of the city, any of whom could have stood a chance against a balrog. However, even the likes of Ecthelion and Glorfindel probably needed to get lucky; even the high king of the Noldor, Fingon the Valiant, had lost out to the lord of balrogs when they faced off during the Nirnaeth Arnoediad. Húrin, the strongest of all human warriors, killed seventy trolls at the Battle of Unnumbered Tears, and he was only captured because he fought so fast and hard that the blade of his battle-axe melted. The greatest warriors among the Men of the Third Age, Húrin's kinsmen Isildur and Aragorn, could never have done that.***
Simply put, although the Company contained some of the most illustrious heroes of the Third Age, by that time there was probably no one native to Middle-earth with the might to face down a balrog. Gandalf (at least after his return) is the second-most-powerful being in Middle-earth, and even for him, the struggle against the balrog is still too much for him to survive. Moreover, even though Gandalf gives Durin's Bane a warning—cryptically identifying himself as a Maia and revealing that he recognizes the balrog's Ainur nature as well—
‘I am a servant of the Secret Fire, wielder of the flame of Anor. You cannot pass. The dark fire will not avail you, flame of Udûn. Go back to the Shadow! You cannot pass.’
Durin's Bane is not fazed and knows that he stands a reasonable chance against Gandalf.
*Or, at least, the power needed to make such things again cannot be harnessed while Morgoth's corruption remains sullying the world. According to the Second Prophecy of Mandos, only when Morgoth is slain in the Battle of Battles and the Great Enemy's taint is expunged in other ways (most notably, via Fëanor's repentance of his own crimes) can the Two Trees be regrown as part of the Second Music, as part of the complete remaking of the world.
**Presumably, other balrogs were slain during the greatest conflicts of the First Age—the War for Sake of the Elves and the War of Wrath—but the tactical details of those conflicts were never recorded in Tolkien's works.
***The first-time reader of The Fellowship of the Ring probably will not appreciate the magnitude of the compliment that Elrond offers to Frodo at the end of "The Council of Elrond," but (although Frodo is not primarily a warrior, but rather a martyr archetype) Elrond's estimation of him very high, ranking Frodo among heroes whose like no longer seems to exist.
‘But it is a heavy burden. So heavy that none could lay it on another. I do not lay it on you. But if you take it freely, I will say that your choice is right; and though all the mighty Elf-friends of old, Hador, and Húrin, and Túrin, and Beren himself were assembled together, your seat should be among them.’