On this point I'd have to ask the question: Are Balrogs immortal? Do they ever die without being slain? If they do live forever, as dragons do not, then they could still be around. Probably sleeping deep in the earth like Durin's Bane was.
Yet I doubt dragons would still be around. There is some convincing evidence that they seriously did exist, but I don't think so anymore. Consider the following:
1. Dragons have not been seen for some time. They could still be in hiding, but even then the youngest would have to be over 700 years old. That's getting up there, even for a dragon. So it is unlikely that they would still be alive.
2. Dragons reproduce sexually. I am under the pretense that dragons were not cohabitational. It would be very difficult for two of them to travel unseen to a place where they could mate. So no new offspring. Also, the new offspring would eventually have to leave their mother's home and find one of their own. They have not been seen doing so.
3. Knowing the nature of dragons to sit on enormous piles of treasure for extensive periods of time, they would not be satisfied to hide in an empty hole. Presumeably, they would be seen flying around looking for a big enough pile of treasure. Yet they have not been seen. This means they must have already found their treasure. The most logical spots for them to find such a treasure would be in Egypt or under Wall Street (National Treasure!
). Since we have not encountered any dragons while excavating these places, it would make sense that dragons are not around anymore.
This brings about only one possible solution: Balrogs could still be around, but dragons are, in fact, extinct.