Dying from the base sounds more like squash vine borers. Yes, they harm watermelons and cukes. Where are you? SVBs are only in the east and south. Of course, Downy Mildew looks a lot different. Those leaves are splotchy. SVB damage just turns leaves brown and they die. It would be handy to see a photo of the dying leaves. Downy Mildew usually affects the uppermost leaves first. Of course, SVB damage usually just collapses the whole vine all at once, but I've seen it go bottoms-up.

The leaves look pretty much like the Downy Mildew photos. (Southeast Montana). The vines die slowly.

Powdery mildew on some plants here is a given. I get it on my pumpkins, cucumbers and gourds. For me the best to use is to spray the leaves with milk diluted with water with a touch of dawn in it. I try to start spraying before the mildew starts. I wonder if it will also work on downy mildew?