Can a family member use my iPhone, or does Touch ID lock them out on the iPhone SE?
You can use your device without enabling TouchID/FaceID.

iPhones (and iPads) are designed for a single user. iOS is unfortunately not multiuser. 

With TouchID, you could train one finger for each person allowed to use the phone, but since there's no concept of separate people, all you'll do is make TouchID less reliable for everyone.

If the phone will only be used at home, just don't enable TouchID and be careful not to store any financial or private information on the phone.