What causes my sister-in-laws cucumbers (new ones) climbing her fence, to...
It sounds like a pollination problem, since it is just the fruit that is turning brown. The plant will abort poorly pollinated fruit, and it sounds like that is what is happening.
Your sister-in-law may want to try hand pollinating any current female flowers to see if that improves the situation.
To hand pollinate, you can either use a paint brush, cotton swab, or the entire male flower. Remove pollen from the center of the male flower using the brush or swab and then tap it on the stigma (center) of the female flower. Female flowers are easily identified because they have an ovary which looks like a miniature version of the mature squash. Male flowers just attach directly to the stem.
This resource has some great pictures to help ID the different flowers and flower parts needed for hand pollination: http://www.missouribotanicalgarden.org/gardens-gardening/your-garden/help-for-the-home-gardener/advice-tips-resources/visual-guides/pollination-of-squash-and-pumpkins.aspx