sounds like a fungus, i was reading on a site earlier trying to find out why mine are jumping off the vines.
it sounds like 'gummy stem blight'
here's the link

Here is a link that might be useful: gourd disease

I do not think it is a problem at all. If the vine is fine other than baby gourds dying everything is okay and it is not gummy stem blight. All that is happening is that for whatever reason the gourds are aborting development and rotting away. This is completely natural since every gourd you see will not develop fully. I have found that to be especially common early in the season before the vines are big enough to support the gourds need for nourishment and also the pollinators (in your case moths) are not in full swing yet.