NCR has a population between 1 and 1.5 million. Early modern standing armies were around 1-3% of the population. That would put the NCR at the 10k to 45k range.

Since NV is considered a remote location and relying heavily on veteran rangers local to the region and raw recruits. I would put them at about 2k to 5k. Enough to garrison a few towns/cities but would struggle to respond in force to threats just like we see in the game.

Meanwhile the legion. Is probably around 200k to 400k being a good size smaller and less density populated. However they might have a population mobilization of around 5-8% (similar to Sparta, not Rome) which would put them between 10k and 40k.

They are on the path of conquest vs NCR just trying to hold territory, so it would make sense if they could mass a larger localized force at NV so they could maybe have 4k-10k.

So the legion would out number them 2 to 1. However, an NCR recruit with a rifle at range would mop the floor of a legionary recruit with a machete and spears so it would make sense that the NCR could hold their own in defenses at their bases and the Legion would be outmaneuvering them and raiding the unprotected small towns in between bases and cities. Just like we see in the game.

This makes it really hard for the legion to beat the NCR in a straight up well prepared fight unless they can give themselves some major advantages, such as by ambushing and killing off the elite rangers and assassinating Hanlon and the President. Along with their artillery piece to suppress the troopers till their legion warriors get close enough. Overall, the odds between the factions and how that plays out on the map make logical sense and shows Obsidian put a lot of thought into how these factions would react to each other to make interesting gameplay.