I really hope no one is storing their financial records and health information on Dropbox...
You can try to get away from it, if you like. But then you're just wasting your time, faxing the same amount of data over immensely slow connections to your bank, who's just going to digitize it anyway and put it god knows where....
And anyone who has done this knows it's ALL time sensitive, and you have other responsibilities in life, so the most convenient/fastest method is the only real way of getting it done.
As far as worst case scenarios go, the financial system does a decent job of making you whole in the event of fraud, and doctors don't rely (solely) upon the stuff in your Dropbox account to make important medical decisions.
That's not to say there's no danger. There's obviously no shortage of Bad Things that could be done with your financial and medical information, but for most people, it's not much worse than the Bad Things that could be done with the photos they share on Facebook, the e-mails they archive in Gmail, or the phone numbers and addresses scattered across countless unsecured databases around the web. In fact, I'd argue that most Dropbox users would be more concerned about Dropbox leaking their private photos and letters than credit card statements and medical bills.
In 2015, the only sensible approach to security is to consider every machine you touch to be compromised, all the time. And yet I still store financial information on Dropbox - pay slips, bank records, mortgage documents, wills. Why would I care? I don't put download links to them on every email I send out of course, but if they would 'leak', it wouldn't make a material difference to my life. While the ease with which I can access and keep track of them does.
Same with medical information - what do I care that anyone knows how strong my prescription glasses are? Or if I'd got some serious disease tomorrow, what sort of treatment I get? I'd have to disclose if I wanted to get life insurance anyway.
I no longer see a rational reason for trying for absolute secrecy on things like this. The cat's out of the bag.