When I went on my RV trip to Alaska, I set up most of my bills to be paid via automatic transfers from my account and paid my mortgage in advance for the months I was gone.
Logging in to your bank, etc. from your computer at, say, a Starbucks is not unsafe provided you don't accept any new trusted certificates from the access point (which reputable places will not even request). The connection to the bank is encrypted on your computer and decrypted at the bank; no one in between can snoop on the contents of the data. (You'll see the little padlock icon in the web browser, and the URL will start with https rather than http.) You should, of course, take care to avoid more basic things like someone standing behind you and watching your keyboard and screen over your shoulder.
You could also log in via a phone using a cellular data connection rather than relying on some Wi-fi access point, if it makes you feel more secure trusting the phone company's connection rather than the coffee purveyor's or campground owner's connection.
