What is the minimum distance between the Caspian Sea and the Black Sea? Is there any water route between them?

836 miles.

There is a proposed canal route, but it is not yet active.

https://en.wikipedia.org/wiki/Eurasia_Canal

I've got it minimum distance at 301.9 miles by the Google Earth Distance measure, and no there is currently no water route connecting the two.

https://earth.google.com/web/@42.74493041,45.93612211,2303.30859576a,1080727.85014182d,35y,0h,0t,0r

https://en.wikipedia.org/wiki/Eurasia_Canal

There seems to be a canal at least part way - but I find https://en.wikipedia.org/wiki/Manych_Ship_Canal rather confusing. It starts off "The Manych Ship Canal is a canal between the Black Sea lagoon the Sea of Azov and the Caspian Sea.", but further on, it appears that it is not complete (or is too small). On the other hand, the Volga-Don Canal does give a more roundabout way from the Black Sea to the Caspian, and this link says that is the shortest navigable route https://en.wikipedia.org/wiki/Volga%E2%80%93Don_Canal which does not clarify the status of the Manych Canal. (63 miles is the distance between the two rivers, not the two seas...)