1. You want people to prove that they're citizens and eligible.
How do you do that? A birth certificate, naturalization document or passport can prove citizenship but many valid citizens don't have them, and many people who do have those documents aren't eligible voters because they've committed crimes. In states that restrict felon voting rights, how can you prove that you've never been convicted of a felony? Maybe in that state it is easy, but in another state? And then there's the issue of proving that you live in the address you say you live in. How do you do that?
This is complicated stuff, but none of this has anything to do with registering online vs. through the mail. There's nothing less secure about registering to vote online.
I agree with you that policing voter eligibility through the registration process is at least difficult but I do not believe, with time, that it is impossible. Voters should be vetted prior to an election, not afterwords. Our current system is rife and open for fraud because that is the way our politicians want it and apparently they are attempting to make vote fraud even easier. As is commonly known, hacking and intrusion of ones privacy is common on the web regardless of the encryption, fail safes and walls so no matter how hard we try, the internet is not secure. It would be hard to convince me that incorporating the internet will somehow make our voting system more secure. Not to preach but our vote lies at the very foundation of our republic’s legitimacy and it needs to be protected at all costs. A government by and for the people depend on it and our government is failing on this issue.