Usually when the little gourds shrivel and die its because they didn't get pollenated.
Try hand pollenating them especially when it rains washes off alot of the pollen off the male flowers,, making pollinationd harder.
Cathy

Thanks Cathy, that's what I thought too. But this rotting is happening to gourds that I did pollinate. It must be something else going on. Then of course, maybe I'm not pollinating correctly. I'll try and look for instructions on how to pollinate right.

I measure the biggest gourd to be 24" around the widest part and 12" tall. I don't know what kind they are - I thought I bought and planted canteen gourd seeds, but obviously that's not what's growing. They're round with a 5" neck. I suppose they're kettle gourds (who knows!)

Oh wow if they are big gourds and not tiny new ones then
you might look for some king of bore or bur or something like that.
Cathy

In case anyone wants to know ... I found a tiny light green, almost transparent worm eating on one of the baby gourds.