Most people simply wash a digital thermometer in soap and hot water to sanitize it between uses. But according to the CDC, this is not how you should be cleaning thermometers. Instead, you need to sanitize a thermometer using rubbing alcohol.
You need to use rubbing alcohol which has at least 60% alcohol in it, so be sure to read the bottle, especially if you are purchasing rubbing alcohol from a dollar store which can carry rubbing alcohol with a lower percentage of alcohol in it.
To sanitize, use either a cotton pad (like a cotton makeup remover pad) or a cotton ball and soak with a liberal amount of rubbing alcohol. Then use it to thoroughly wipe down the entire thermometer. You want to pay special attention to areas at the top and around any buttons or other crevices where bacteria could get caught.
Allow the thermometer to air dry. Do not wipe it dry, as you want the alcohol to do its sanitizing as it dries. After it has air dried, you can rinse it with water to remove all traces of alcohol, being careful to not soak the digital reader area.
Never store a thermometer in a case unless it is completely dried after sanitizing.
You cam also get plastic sleeves that fit on many types of thermometers, as an extra precaution to prevent transferring germs accidentally.