Battlestar Galactica's final survivor count was much lower in the series finale due to the Second Cylon War - and partially due to humans fighting themselves - but there were still enough people alive to ensure the race survived. Beginning with the destruction of the Twelve Colonies, dozens of ships escaped the Cylon invasion and banded together with the Battlestar Galactica and formed the Fleet - all that remained of the human race - and together they all sought out the location of Earth.
It wasn't an easy journey, of course, and thousands of people lost their lives along the way. President Roslin first estimated the number of survivors in the Fleet to be 50,298, which was about one week after the initial attack by the Cylons. Unfortunately, that number began to drop quickly in the following weeks, beginning with the destruction of the Olympic Carrier - a ship that was purposefully destroyed by Galactica's pilots after they identified a nuclear signature aboard, fearing that it was a threat to the Fleet.
Approximately four years later, the final survivor count numbered 39,406 people at the start of the Battlestar Galactica series finale, "Daybreak, Part III". However, upon reaching the second Earth - our Earth, not the planet the Thirteenth Tribe had originally settled on after leaving Kobol - Admiral Adama noted that the number of survivors totaled about 38,000. It could've been a miscalculation on his part or more than a thousand people were killed at some point after the Battle of the Colony.
Regardless of the reason, going from 50,000 people to 38,000 after four years on the run and countless battles with an advanced cyborg race is still an achievement, especially considering Gaius Baltar estimated the human race would've gone extinct in 18 years had they not found a new home. But much of that drop could've been avoided if they didn't listen to Baltar when he decided to settle on New Caprica, in the first place. A year after they landed on the planet, the Cylons arrived and occupied them, ultimately resulting in about 10,000 deaths in just a few months.
The Fleet was never able to recover from that, and matters were only worsened after the Battle of the Ionian Nebula, in which multiple ships were either damaged or destroyed by the Cylons in a surprise attack. From then on, the Fleet's population began dropping at a steady rate - in many cases due to infighting and suicide after leaving the first Earth - until they reached the new planet. The roughly 38,000 survivors who made it to Earth were able to keep the human race alive, eventually seeing that number grow to more than 6.7 billion after over 150,000 years (the then-current population of Earth in 2009).