The vine (any vine, for that matter) needs GREAT air circulation and sunlight. I would recommend thinning the vine to provide better air circulation and not using any fertilizer. This should take care of the problem.

Start cutting and thinning near the ground and place the cuttings in a garbage bag, then toss into the trash. This vine gets massive and you will need to keep the vine on the thin side all the time.

Evan,
I lightly sprayed rose spray on the vine in early spring before it really started growing good, do it twice a month til it really starts getting warm where you live, probably the end of May.
It really works and won't hurt the vine at all.
GOod Luck!

Thanks for the comments. Unfortunately I have excellent air circulation, lots of sun, have sprayed aggressively, and still no luck. I will try spraying as early as possible this year and see if I can get ahead of the problem.

I'm afraid the real problem is living in NJ: we had so much rain this year that everything that could possibly have problems, did so. My verbena bonariensis was covered with mildew even in absolute full sun. The phlox was a disaster as well.

The situation is so bad that I am taking out some of my David Austin roses and replacing them with Kordes disease resistant roses. I'll give the honeysuckle one more season but I think that it is going to have to come out and be replaced with sweet autumn clematis which doesn't seem to be bothered by the dampness.