I've been online-only for many years now. It breaks my heart to think of people paying "account maintenance fees" at the big banks.
The main problem I ran into was depositing checks. I used to actually have to mail a check to the bank (although, they gave me pre-paid envelopes.) But this is largely a thing of the past due to mobile apps that let you deposit a check by taking a photo of it.
Other things to consider:
* Getting cash: this is the most important, so find out about it before you open an account. There are three basic ways to do it: 1. Get a bank with an extensive ATM network; 2. Get a bank with ATM fee rebates; or 3. Get cash back from grocery purchases. This last requires that you buy something every time you need cash, and is often limited to $40 or so.
* Depositing cash: I haven't found a good way to deal with this, other than just avoiding getting large amounts of cash. If, for some reason, you find yourself with a wad too big to carry around, the best I can come up with is to store it in your sock drawer, and then just use it instead of the ATM until it's depleted.
* Getting a roll of quarters: You have to visit the local video game arcade, because your online bank is not going to help here.
* Transferring money: From now on, using ACH transfers from the website is how its done. This takes a few days to set up, so set it up before you need it. Be aware that some banks have a fee for outgoing transfers. This means that you should always make the transfer at the receiving bank.
I would recommend: CapitalOne, Discover, Ally. CapitalOne, has free ATMs in Mexico, with no exchange fees!
I would not recommend: Emigrant Bank. It's a fine bank, with a great interest rate, but it's limited in the ability to transfer funds, and has a clunky website that suggests it doesn't quite have all it's s*** together.
There's also an online bank called "Redneck Bank", which I've never used, but people probably get it just for the name on the debit card.