I'd love to know if they have any idea on the numbers
As the Eldar are very secretive about this and the empire itself has a lack of knowledge in several areas there (or sometimes it is so that specific individuals or organizations think the empire doesn't need to know some things and thus keep that exact knowledge under the hood): The empire doesn't know how many eldars really exist.
In canon there is no concrete number ever mentioned as far as I'm aware. Additionally the above secrecy problem is also there. BUT what the empire DOES know is that the Eldar live on world ships (Ulthwe, ...) and have some maiden planets. And they had quite a few encounters (as friends and enemies alike) with a couple of thousand eldars each at the very max (mostly a few dozen to hundred). And in addition that the Eldar are a fallen race. So it is likely that the empire estimates the number of Eldar in the millions or low billions. But that will always be only a very rough estimation as they really have no data on that (at least officially that is).
One last point though:
- From the view of the empire though regardless how many exist....they are way too many. 0 Eldars would be the exact number the empire would like to know to exist. As they are just vile aliens and nothing else.
Enough as are required.
It's the same for any race in 40k or WHF, GW can't and won't release official figures like an actual number of living Eldar. GW keep things vague on purpose, they want you to fill in the gaps yourself, build your own armies and create your own lore.
If you're reading the fluff, from one book to another you can expect the number of living Eldar to vary dramatically. There are always enough Eldar to move the story forward while somehow remaining on the brink of extinction.
This reddit post is an interesting read.
Since there is no better answer than my comment, let me use it:
There never was any official statement from Games Workshop so its a bit of case "depends of the author": In early fluff there were only dozens craftwords with few dozen millions each. Later this was retconned into thousand craftwords each housing up to half a billion. Add to that Exodites that live like wood elves and Dark Eldars that keep cloning themselves like crazy and you have the "not-so-dying race"