I invited my aunt to share my family tree, after she had helped me with some details, she has gone into it deleted all her details, her children, partners etc and emailed me an angry message about not wanting her details on the internet!! I am replied assuring her it is private, and changed the settings so now only I can edit it. Obviously I want to add all these deleted details back on but she is asking me not to.
I've run into similar problems in the past. As we know, basically anyone over the age of 30 is just now within the last 10 years getting used to the deprivitization of the internet. The one I heard a couple times from my mom (who doesn't use credit cards on the internet so that she can't have her identity stolen) is "everyone's banks and accounts always ask for your mother's maiden name when verifying you - if we put that information here, people will be able to assume our identities!"
There's no real way to sooth or sway this opinion. The best thing you can do is make sure your privacy settings do not allow outsiders to access your tree's details (and test it by logging out yourself and then searching for your tree to see what it'd look like for an internet surfer to find your tree), and then have your aunt do the same. In addition, let her know this isn't the worst place as far as private info. Most state and local offices here in the states will cough up personal info (sometimes even as bad as SSN and driver's license) just from a phone call.
Assure your aunt that she's damned if she does and she's damned if she doesn't...so she might as well join the site!