The cucumbers bloom and get a small cucumber on them and they die when they are about 1 and 1/2 inches long. What is wrong?
There's a good chance that you're lacking nutrients in your soil, if the leaves all look good and there's no sign of pests. Don't forget that crops should be rotated each year, too, which helps replenish soil content that's vital to your plants. Then again, if this has only happened to the first few cucumbers on the vine, your plant may simply be struggling to get going this year, and with continued proper watering and light or some slight adjustments, the later cucumbers could mature in a more acceptable fashion.
These articles may help you identify what the issue is: