If you want to check on a bank’s FDIC insurance, the FDIC website has a search page where you can check it out.
If the bank is insured by the FDIC, your money’s safe (up to $100k), so how “good” the bank is doesn’t make much difference in that respect.
As much as I like using an Internet bank, there are some downsides. If you frequently hit an ATM for cash, you’ll probably have to pay fees to whomever owns the ATM you’re using - your Internet bank probably won’t have any ATMs of its own, or if it does, it’ll only have a few in its home territory (Presidential, for example, has only a handful in MD and NoVA, one within walking distance of my house, thankfully).
If the online bank doesn’t have any ATMs or branches near you, there is literally no way to deposit cash into the account. You can have your paychecks electronically deposited, and you can mail in any other checks you get, but if someone gives you a fist full of c-notes, you’ll have to sell them to a friend (who will deposit them in his account, and pay you for them with his own check, which you will then mail to your online bank).
It can also be difficult or impossible to get a cashier’s check or certified check if there’s no branch office near you.
The upsides include very good rates, free billpayer service (from Presidential, at least), and good, reliable electronic access to everything.
One possible approach: maintain two checking accounts. Keep the bulk of your vast fortune in the online bank, for the rates, and keep the bare minimum in a local bank, to preserve access to a free ATM, to let you deposit cash, etc. Then, just write yourself the occasional check to move money from one account to the other.