If you want to keep whole cucumbers fresh after picking them take some saran wrap or other plastic wrap and wrap the whole cucumber in plastic wrap which will keep the moisture out and help the cucumber stay fresh longer.
Then after wrapping the cucumber in the plastic wrap place the cucumbers into the refrigerator and they should keep for up to a week.
You can also keep cut cucumbers fresh by placing them into a container and then cover the cut cucumbers with water and then cover the container and place them into the refrigerator as well and they should also keep for up to a week.