Hi Scratch,
A friend of mine has just mentioned that there is a letter in this months Trout & Salmon, which is written by someone who has fished the Border Esk for many years, in the letter it states how badly the river has declined and how poor the fishing is, so you may want to pick up a copy!
If you want to fish the river, then one of the most popular areas is Langholm
http://www.trout-salmon-fishing.com/...border-esk.htm
Depending on what you are fishing for the Annan may be worth considering?
http://www.fishannan.co.uk/index.asp?dom=FishAnnan
http://www.riverannan.co.uk/
Hope that helps!