Use baby wipes in their container, don't take them out, leave them flat right in the container. When you need to clean the ink off the rubber stamp, just open the lid and stamp, stamp, stamp. Then dry on an old towel. Remember, never rub the image on the rubber stamp as that ruins it. You can use each of the baby wipes several times. If they last so long as to dry out, just dampen with water. I have done this for years, and my rubber stamps are as good as new; just a little stained around the wood mount.
I have used baby wipes for years to clean my rubber stamps. I have rubbed the rubber image and have never had one damaged. If you find the wipes dried out, use distilled water to refresh them. Tap water will lead to mold.