From what I've read, online only banks are safe as long as they are FDIC insured. It's just like how you use the app or website for your regular bank - just without that brick-and-mortar overhead.
One thing to remember though: *personal* online security is still up to you. (aka maybe don't use your banking apps when you're on the free coffee shop WiFi.) The bank will take security measures on its end so you can use the app or website safely, but you still need to practice good online habits.
I have a business checking account with an online only bank and so far I've had a great experience.