Perhaps I'm not qualified to answer, since I've never tried putting gutters on the fence. It does seem to me, though, that unless you have a large collecting area (such as a roof) connected to your gutters, you wouldn't be gaining much (if I understand the scheme).

Are there plants/shrubs in that garden which demand more water than the average rainfall in your area? If so, would you consider replanting with more drought tolerant selections? Another idea: can the slope be "terraced," which would improve water retention in the beds (and decrease erosion)?

Part of our yard is on a steep slope, so I sympathize. I don't want to terrace it, however, because in the winter it makes a wicked sledding hill :-)!

You might want to check out "swales" in permaculture books. I don't know how steep your property is, but swales slow down & hold whatever water falls.

One good book is "Gaia's Garden" -- explains it quite well. Often in libraries.

'Pup

In a book I read about sloping gardens all the plants had a small semicircular bank of soil in front of them, the water coming down the slope would dam in this and the plant would get water instead of it rushing past it. Is this a swale?

I showed my dh a spot in our garden where I thought we could put a rainwater tank, he said that there's nowhere for the water to come from. I suggested running guttering along the top of the fenceline, he thought it was a stupid idea.