Don't worry. It is not a big deal as long as you wash your hands with soaps carefully before putting on your contacts. After putting on your contacts, you can apply some artificial tears to flush the contacts and this will help clean the fingerprints. Or the fingerprints will go away after blinking your eyes for a moment because the water in our eyes can help clean it. Besides, the fingerprints will not affect your vision. So you really needn't worry about it.
If you are using the hard contact lenses, you may get fingerprints on the surface of your lenses. But if you choose the soft contact lenses, you will not get fingerprints on the lenses. So I think you can change your contacts if you want to avoid this problem. And you must use your fingers to put on your contacts because there isn't other ways to put on the contacts.
Our eyelids will work like the windshield wipers on a car. So they can help clean the fingerprints on the surfaces of the lenses after we put on contacts. But I really doubt that whether the fingerprints you saw are really fingerprints. Or what you saw was just the solutions on the contact lenses. The solutions on the contacts will wash away your fingerprints too. All in all, I think they will not do great damages to your eyes as long as you wash your hands clean every time you put on your contacts. Good luck!
You may need to buy a pair of good contacts because usually only the cheap contacts will have fingerprint marks while good quality lenses will not leave behind any fingerprint marks on the surface of the lenses. And good quality contacts will make your eyes feel more comfortable. Hope this helped!
Do use fresh, contact lens cleaning solution every time. Don't use tap or sterile water, saliva, saline solution or rewetting drops. None of these serve to disinfect and properly clean your contact lenses. Do rub your contact lenses with your fingers, and rinse them with fresh cleaning solution afterward.