There are any number of ways to do this, but the easiest is to simply use a separate layout for printing that doesn't include that field.
We scan driver's license Mag strips here where I am and we scan the info into a 2 pixel field to conceal the scanned data from the user--which is helpful when you need the field to be part of the layout, but don't want the user to be able to see the contents of the field.
If you don't mind the info being visibly stored on the layout but don't want it printed, you can select the field in edit mode and set it to not print.
You can also put a "shield" on the field. create an object that is linked to a script which only allows someone with adequate privileges to see the field. If you put a shield, select a fill color for the shield so it isn't see-thru. The script can bring up a dialog with the information or it can bring you to a separate layout where the information is displayed as view only or editable.
Thank you these were both very helpful!