The steps in Integrated Pest Management are,
1. Identify the problem, (which you have done)
2. Determine if a control is necessary. (also done)
3. Determine the least toxic, least environmentally harmful, means of control.
4. Apply that means of control.
5. Review.
So you have a problem, the Stripped Cucumber Beetle, in fairly large numbers and since they are on the plants Floating Row covers would not help at this time. Next would be are there predators present in large enough numebrs to control the buggers and if not what would be next? Would Insecticidal Soap work? If that did not then maybe Neem Oil products. If that did not work then maybe a pyrethrin product. And for next year plan on placing the Floating Row Covers when the melons are planted as well as making sure you have a good, healthy soil to grow them in.

OP wrote "there was a SWARM of striped cucumber beetles, or at least a yellow striped flying beetle that looked like it."

Cucumber beetles normally hide at night. So, it's wise to determine what you have at night versus what's eating your melon plants.

Can you post pictures?

If not, please describe size, shape and precise markings.

Beyond that, you can secure some of the beasts in a clear container, then take them plus damaged leaves to a knowledgeable source -- perhaps large independent garden center -- for suggestions.

Do the two fit? That is, did the captured critters make the damage? If not, what did? And then what to do.

If it's not cucumber beetles, it could be potato beetles.

For melon-pests in general, I've found that clearing the weeds away from the base of the plants, and then spreading diatomaceous earth on the leaves and around the base helps a good amount... until it rains and gets washed off/clumped up. And then they'll come back.

It is definitely striped cucumber beetles. I've trapped the bugs and compared them to pictures. I went out at 2 am and checked my squash and melons and found tons of them climbing all over the leaves and flowers and all over the ground before they flew away.

I'm thinking of putting some ant/pyrethrin and DE dust around the base, but the infestation is pretty bad and I doubt it will be of much help

I will try to post a picture if it is necessary

I'd hit the leaves, too. The reduced photosynthesis will be made up for by the bug slaughter ;)

Pyrethrin's a pretty good insecticide for a broad spectrum -- just keep it away from the flowers, since it can hurt bees. It doesn't kill a number of types of predators (although it does kill some), and it gets most pests. It leaves earthworms alone, too. I once soaked a pot in a strong pyrethrin solution to kill of an infestation of root-eating pests, and a week later, was transplanting some plants from that pot and found that the pot had a number of large earthworms in it, still perfectly happy.

I have some sort of pest that seems to attack the vines - not the leaves - no bored holes, but a brownish narrowing of the vine, often a foot or two nearer the root from a fat melon, so that the distal part of the vine with any melons dies. Anyone know what does this? Thanks/ MC

Marvin, you would have been better off starting a new thread rather then tacking onto one two years old.
However, what you describe sounds like it could be the Squash Vine Borer.