I signed on with one of the first Internet-only banks, ING Direct, because it looked like a neat idea and I wanted to test-drive it. ING Direct has since been acquired by Capital One. I've been with that for at least 20 years, and have had zero issues where I would need to visit a branch, or even speak with someone on the phone. ATMs, website, and phone app handles all transactions.
On those rare occasions where I have more cash on hand than I would normally have, it gets taken care of by a trip to the grocery store for a few items, and change goes in the self-checkout register (doesn't cost anything like CoinStar does).
I personally am leery of online banking because I feel it is too easy to hacked; but, that is just me.