As with many issues in Tolkien's mythology the answer to this question depends somewhat on which texts we look at. Conventional wisdom holds that the 'correct' answer is that there were only a handful of Balrogs. However, an analysis of the books shows that this idea was in fact much later developed and less firmly entrenched than might be commonly supposed.
The earliest texts of the mythology show Balrogs in great numbers;
"Melko sent his host of Balrogs after them, and Mablon the Ilkorin died to save them when pursued."
BoLT1, Gilfanon's Tale
"Some were all of iron so cunningly linked that they might flow like slow rivers of metal or coil themselves around and above all obstacles before them, ... and upon them rode the Balrogs in hundreds; and these were the most dire of all those monsters which Melko devised against Gondolin."
BoLT2, The Fall of Gondolin
"The early conception of Balrogs makes them less terrible, and certainly more destructible, than they afterwards became: they existed in 'hundreds' (p. 170),* and were slain by Tuor and the Gondothlim in large numbers: thus five fell before Tuor's great axe Dramborleg, three before Ecthelion's sword, and two score were slain by the warrior's of the king's house."
BoLT2, Commentary by CT on The Fall of Gondolin
This idea of numerous Balrogs was also carried forward through years of new stories and revisions;
"Wherefore each embassy came in far greater force then they had sworn, but Morgoth brought the greater, and they were Balrogs."
SoME, The Quenta - 8
"There came afresh a hundred thousand Orcs and a thousand Balrogs, and in the forefront came Glomund the Dragon, and Elves and Men withered before him."
SoME, The Earliest Annals of Beleriand - Year 172
"Wherefore each embassy came with greater force than was agreed, but Morgoth sent the greater and they were Balrogs."
LROW, Quenta Silmarillion - Ch. 8 ~89
"But at length after the fall of Fingolfin, which is told hereafter, Sauron came against Orodreth, the warden of the tower, with a host of Balrogs."
LROW, Quenta Silmarillion - Ch. 11 ~143
"There came wolves and serpents and there came Balrogs one thousand, and there came Glomund the Father of Dragons."
LROW, Quenta Silmarillion - Ch. 16 ~15
So where does the idea that there were only a handful of Balrogs come from? It can be traced to a footnote Christopher included attached to his commentary quoted above from the Second Book of Lost Tales;
"* The idea that Morgoth disposed of a 'host' of Balrogs endured long, but in a late note my father said that only very few ever existed - 'at most seven'."
BoLT2, The Fall of Gondolin
This seemingly direct statement, published in 1984, shaped the common idea that Balrogs were few in number. It was also often assumed that though the idea of many Balrogs had "endured long" it must have been discarded prior to the writing of LotR. In fact, the Balrog hosts persisted even after the completion of The Lord of the Rings. They can still be found in the Grey Annals and Annals of Aman, both of which can be dated to sometime in the 1950s;
"There came wolves, and wolfriders, and there came Balrogs a thousand, and there came worms and drakes, and Glaurung, Father of Dragons."
WotJ, The Grey Annals - Year 472 ~230
"It came to pass that at last the gates of Utumno were broken and its halls unroofed, and Melkor took refuge in the uttermost pit. Thence, seeing that all was lost (for that time), he sent forth on a sudden a host of Balrogs, the last of his servants that remained..."
MR, The Annals of Aman - 1099
What is more, there is a note attached to this Annals of Aman text... which is almost certainly the same passage to which Christopher had referred in BoLT2;
"In the margin my father wrote: 'There should not be supposed more than say 3 or at most 7 ever existed.'"
MR, Section 2 (AAm*) - note 50 just before Section 3
So, this then was the 'turning point'. Yet there
is no way of knowing whether this idea of a limited number of Balrogs
would have been retained. Given the difficulty in precisely dating
these texts it is even possible that the Grey Annals (GA2) reference
to 'Balrogs a thousand' post-dated the Annals of Aman note saying
there were 'at most seven'. We thus have the possibly unique
situation of a widely accepted point of Tolkien lore which is
contradicted by every extant narrative writing on the subject, and
indeed was only ever found in a single post-LotR marginal note.
Another 'angle of attack' we might use in deriving the total number of Balrogs might be to count how many were slain in the First Age and add the number said to have survived thereafter. However, while there are references for these figures they too changed over time. At first Tolkien indicated that the Balrogs were all destroyed;
"The march of Fionwe into the North is then told, and of the Terrible or Last Battle. The Balrogs are all destroyed, and the Orcs destroyed or scattered."
SoME, The Earliest 'Silmarillion' ~18
"But Fionwe crossed Sirion and the hosts of Morgoth were driven as leaves, and the Balrogs were utterly destroyed..."
LROW, The Later Annals of Beleriand ~350 
However, sometime around 1937 Tolkien changed this to;
"The Balrogs were destroyed, save some few that fled and hid themselves in caverns inaccessible at the roots of the earth."
LROW, Quenta Silmarillion - Conclusion ~16
It is interesting to note that this was several years before Tolkien wrote of the Balrog in Moria, and it seems unlikely that Tolkien had one in mind for that scene as he originally wrote Gandalf's opponent as one of the Nazgul;
"They are pursued by goblins and a B[lack] R[ider] [written above: a Balrog] after escaping from Balin's Tomb - they come to a bridge of slender stone over a gulf.
Gandalf turns back and holds off [?enemy], they cross the bridge but the B[lack] R[ider] leaps forward and wrestles with Gandalf. The bridge cracks under them and the last they see is Gandalf falling into the pit with the B[lack] R[ider]. There is a flash of fire and blue light up from abyss."
RotS, The Mines of Moria
Thus, the reference to Balrogs surviving at the roots of the earth did not originate to explain the Moria Balrog (rather the inverse) and might indicate that Tolkien retained the idea of multiple survivors... as is also suggested by his original ideas as to where the Balrog of Moria came from;
"'A Balrog!' said Keleborn. 'Not since the Elder Days have I heard that a Balrog was loose upon the world. Some we have thought are perhaps hidden in Mordor [?or] near the Mountain of Fire, but naught has been seen of them since the Great Battle and the fall of Thangorodrim."
This again implies that there might be more than one surviving Balrog. However, this and the other references pre-date the idea of changing the OVERALL number of Balrogs to 'at most seven'. It would seem difficult to maintain that figure given the references above to Balrogs (plural) being slain in the fall of Thangorodrim and various more detailed accounts of those slain in BoLT2, The Fall of Gondolin;
"...but the men of Rog leapt upon the coils of the serpents and came at those Balrogs and smote them grievously, for all they had whips of flame and claws of steel, and were in stature very great. They battered them into nought, or catching at their whips wielded these against them, that they tore them even as they had aforetime torn the Gnomes; and the number of Balrogs that perished was a marvel and a dread to the hosts of Melko, for ere that day never had any of the Balrogs been slain by the hand of Elves or Men. ... Fearful too they were for that slaughter Rog had done amid the Balrogs..."
The precise number slain is not given, but obviously a significant number. The other references all give exact amounts;
"...and by reason of the great doughtiness of those two lords they came even unto the Balrogs. Of those demons of power Ecthelion slew three, for the brightness of his sword cleft the iron of them and did hurt to their fire, and they writhed; yet of the leap of that axe Dramborleg that was swung by the hand of Tuor were they still more afraid, for it sang like the rush of eagle's wings in the air and took death as it fell, and five of them went down before it."
"There seeing the wavering of the enemy by reason of the dread of the fall of Gothmog the marshal of the hosts, the royal house laid on and the king came down in splendour among them and hewed with them, that they swept again much of the square, and of the Balrogs slew even two score..."
In addition to the twenty-eight named above there were also Gothmog, slain by Ecthelion, and another Balrog slain by Glorfindel... coming to a total of 30 plus however many slain by the men of Rog, however many slain during the fall of Thangorodrim, and the Moria Balrog and any other survivors (35 minimum). Clearly far too many for the later figure of 'at most seven'. Tolkien would thus have needed to revise his ideas such that fewer were explicitly killed, ONLY the Moria Balrog survived, and/or there were some number more than seven originally.
How many Balrogs were there? A case can be made for three, seven, thirty-five, hundreds and thousands... but it is clear that they remained numerous from the earliest days of the mythology through the writing of The Lord of the Rings. Thereafter the answer must remain uncertain as even the 'last word' cannot be precisely determined.