About twice the height of an Elf
Depending on you views of the canonicity of the Book of Lost Tales - they are basically the draft notes for the Silmarillion.
(Personally, if they are not contradicted by later canon I tend to take them at face value)
Assuming that Balrogs are the same size the Balrogs at the Fall of Gondolin appeared to be of a consistent double elf height.
Then sprang the Balrog in the torment of his pain and fear full at Glorfindel, who stabbed like a dart of a snake; but he found only a shoulder, and was grappled, and they swayed to a fall upon the crag-top. Then Glorfindel's left hand sought a dirk, and this he thrust up that it pierced the Balrog's belly nigh his own face (for that demon was double his stature)
- The Book of Lost Tales - Part II - Gondolin
Also from the Book of Lost Tales, when talking about the Fall of Gondolin:
Then leapt Ecthelion lord of the Fountain, fairest of the Noldoli, full at Gothmog even as he raised his whip, and his helm that had a spike upon it he drave into that evil breast, and he twined his legs about his foeman's thighs; and the Balrog yelled and fell forward; but those two dropped into the basin of the king's fountain which was very deep. There found that creature his bane; and Ecthelion sank steel-laden into the depths, and so perished the lord of the Fountain after fiery battle in cool waters.
- The Book of Lost Tales - Part II - Gondolin
Gothmog was obviously of a height that a severely injured Ecthelion was able to drive his spiked helm into his chest. Too much bigger than the elf and this wouldn't have been possible. Gothmog was also Lord of Balrogs so it could be expected and assumed that he would be one of, if not the, biggest of the Balrogs.
If the Balrogs were too big one would not expect Elves, nor a man sized Gandalf (though Maiar powers could be at play here), to be able to go toe to toe with them on numerous occasions.
For example, Fëanor was surrounded by Balrogs until he was eventually worn down:
and there issued from Angband Balrogs to aid them. There upon the confines of Dor Daedeloth, the land of Morgoth, Fëanor was surrounded, with few friends about him. Long he fought on, and undismayed, though he was wrapped in fire and wounded with many wounds; but at the last he was smitten to the ground by Gothmog, Lord of Balrogs, whom Ecthelion after slew in Gondolin.
- The Silmarillion - Of the Return of the Noldor
The Balrogs were then driven off by Fëanor's sons.
Fingon was holding his own against Gothmog until he was assaulted from behind:
At last Fingon stood alone with his guard dead
about him; and he fought with Gothmog, until another Balrog came behind and cast a thong of fire about him. Then Gothmog hewed him with his black axe, and a white flame sprang up from the helm of Fingon as it was cloven.
- The Silmarillion - Of the Fifth Battle: Nirnaeth Arnoediad
Then you have Ecthelion and Glorfindel killing a Balrog each in the defence of Gondolin at the cost of both of their lives.
Admittedly Elves in those days were greater and capable of great feats, look at Fingolfin fighting Morgoth and wounding him several times, but there must still be a close enough difference in size for the combat to be realistic – otherwise the elves would have been overwhelmed by sheer brute force.
Possibly of interest
User Balfrog over at The Lord of The Rings Fanatics Plaza has worked out the Balrog's size based on the Bridge of Khazad-Dûm. I will try to summarise his calculations here in case of link death, however I am in no way trying to take ANY credit for this.
Gandalf stood in the middle of the 50 foot bridge, therefore he is 25 foot from the end of the bridge.
The Balrog stops before stepping onto the bridge, putting it 25 foot from Gandalf.
The Balrog steps onto the Bridge before swinging his sword at Gandalf (user Balfrog argues that this is just one step).
Therefore, the step had to have been half the bridge span minus the reach of the Balrog and its sword's (the reach was assumed to be the Balrogs height)
This gives the height of 16.7 foot. There would of course be a possibility of error based on size of the Balrog's sword, if they had to do a full stride etc.; adding 20% error either way gives a range between 13.36 foot and 20.04 foot.
Which puts it at somewhere between 2 and 3 times the size of an elf if we assume the elf average height to be around 6 foot.