There are several known major Earth colonies, and perhaps dozens (or more) colonies of humans who were not officially affiliated with Earth. There are probably tens of thousands of Starfleet vessels, and if what we've seen is any guide, the vast majority of the individuals serving are Human.
So roughly how many humans, are there, total, in the Star Trek universe? (Yes, I know the Star Trek universe covers several hundred years.) Tens of billions? Hundreds? Trillions?
As James wrote, Earth has 9 billion in First Contact.
There might have been a slight baby boom after WWIII, but developed societies tend to move towards negative population growth, so perhaps Earth's population reaches 10 billion before this happens.
On Mars, the next largest human population, there are 133.8 million (ST: Star Charts), and Luna has 50.2 million (ST: Star Charts and First Contact) by the late 24th century. That adds up to 10.184 billion.
Many other human colonies were similarly wiped out or reduced to only a dozen or perhaps a few dozen colonists by the time they were contacted by Starfleet.
Other non-destroyed human colonies mentioned but whose populations are unknown:
Then there are of course penal colonies and mining colonies, but these generally don't have very significant populations. So if you add up the confirmed population figures, that only totals about 10.2 billion. We can be generous and add another billion for the colonies with unknown populations (though, given the average figures, they're likely to total to only 100 million at most), so that gives us 11.2 billion.
The Federation has 700 colonies (website for Star Trek: TMP) by the 23rd century, and that's split between 120 worlds. Earth, we know, is one of the younger (in terms of space-faring history) species, though we're portrayed as having a greater spirit of exploration. Given all of this, and considering the small (<1 million) populations on most colonies, it's hard to imagine there being more than 12 billion humans by the 24th century.
We know that a Federation census reported 985 billion individuals in 2370. Assuming all individuals are member species (of which there should be around 183, assuming one species per member) and not Federation affiliates (of which there are 7128), then that averages out to 5.38 billion individuals per species.
So I think the 12 billion uppper limit by the 24th century is reasonable, assuming that we're not an especially populous or prolific species. (The census was apparently from the Star Trek: Star Charts, not a canonical source.)
Compare with Earth in the original timeline:
Borg drones have no need for recreation, personal space, or other quality of life considerations. Their machine-like efficiency, spartan living requirements and complete disregard for the environment enable them to completely colonize the planet surface. Therefore, if there are only 9 billion life signs on the surface, we can assume this is the approximate upper limit of humanoid life that the planet can support.
The only figure we have on the Earth population in the regular timeline is the Star Trek: Star Charts 2370 census which found 4.2 billion humans on Earth at that time. Even though this isn't canon, it's reasonable to assume that there probably isn't much more than 9 billion humans on Earth in the 24th century.
There aren't trillions, as the Tkon Empire, with a population in the trillions, was considered huge, considerably larger than the Federation with vastly superior technology. I would assume that the population of the Federation is in the trillions, but that no single species approaches that number.
A few non-canon sources - such as several of the draft screenplays for The Motion Picture - mention the population of Earth, but the total varies. The only canon source that mentions Earth's population is First Contact, which states it at 9 billion. We can probably safely assume that no single human colony has as many people as the homeworld. While we can't come up with a clear total from that, I feel that it is safe to assume a population of "tens of billions" for humans, both in the Federation and outside of it.
