My opinion: FDIC-insured online banks are fine; I use ING Direct. I do not think it is prudent to have an online account as your only bank account, because there are too many things you need occasionally from banks that you might not yet have encountered.
Like others, I have consistently and uniformly, over um one, two, three, four decades, found small local banks and credit unions to be just plain better than big banks. The only thing to watch out for is to make sure that they do offer the full range of "brick-and-mortar" bank services.
a) Medallion signature guarantees
b) Certified checks and/or teller's checks. The need can crop up very quickly. If you don't have a physical bank with money in the account and your signature on file, and you need a teller's check immediately, what do you do?
c) Notarization. There are many other options besides the bank, but do you know what yours is? Last time I needed one, it was because the TSA said I had "a similar name to that of someone on the no-fly list" and every darn time we flew, the wife and kids would get their boarding passes and MINE would have "SEE AGENT" stamped on it and it would take me an extra ten minutes at the counter while they made calls and took my ID backstage and did mysterious things. So I knuckled under and joined the TSA's "this-isn't-a-clearance-but-give-us-all-your-ID-anyway" program, and I needed a notarization, and being retired could no longer get it at work.
d) Foreign exchange. Needed a wire transfer to put down a deposit for a reservation at a fancy family campground in Bergen-op-Zoom. Not sure why a credit card didn't work, but a wire transfer was what they wanted.
e) Safe deposit box.
f) Empty the change jar, without a ripoff 8% charge at the supermarket.
g) Someone to beg to sponsor the Little League team.
Paper checks are already starting to feel like slide rules and carbon paper, but it is amazing how indispensable they still are. Everyone keeps trying the PayPal and smartphone-to-smartphone-zap stuff but there's a long, long tail on the need for paper checks. For example, the firm I consult for does not do direct deposit and pays via paper check (!). And the online banks don't have deposits licked yet. ING's image deposit seems to work but is limited to $3,000. Deposits via ATM machine are very spotty--at one time ING claimed to have it but only for their "Electric Orange" account--and when I tried it, it didn't work. The ATM machine belonged to the NETWORK they said worked, but it turned out that the closest MACHINE in the network that could actually accept a deposit was twenty miles away.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.