FWIW, I'm not "afraid" of giving Disney my fingerprints.
Then you have nothing to fear at all because it's not fingerprints that are taken when you place your finger on the scanner. It's a scan, not a reader.
As for my family? My fingerprints are on record somewhere because I was in the ROTC in college, my husband's are in somewhere because he served in the military for 12 years, my son's are in there somewhere because we had that done when he was a child and that was to protect him in case he was ever taken from us.
I'm 52, my husband is 54, our son is 32. In all that time, we've had a fingerprint match come back to us - 0 times. 0.