Given a "Could file sharing" service, is it feasible to provide non-secure (http) service for free, whereas if you need to store and transfer important data, you pay for secure service (https) a nominal ($5) monthly fee.
Would this be accepted by the users?
If you answer is "NO", please read my explanation:
A community-like initial response would be - no, you can't monetize of security, when it come to file sharing. But think about it - what are the chances that your traffic is getting sniffed on the way to ISP? Low. From ISP to the destination server data center - almost none. So when you are sending someone a photos from a party, you hardly need SSL. But if a designer is sending sketches of a web site, SSL is required when there are competitors, etc.
I believe security can be monetized, and that people will pay for excellent security. You will have to offer far more than just SSL (https) for uploads and downloads however. As just one example of what people might want consider audit-ability of who downloaded the file?
If all I want to do is transmit something securely, why wouldn't I simply encrypt it rather than transmitting it in the clear?
You can monetize anything you want, if you know how to sell it to the customer. In the 80's a computer company sold millions of copies of "RAM Doubling" software, that actually did nothing. Clever marketing let them get away with it until they were finally caught by the press.
If you can convince your customers that secure is better, you can sell it to them.
You do realize that bandwidth is not free, though? You are going to pay virtually the same amount of money for bandwidth for your free customers as for your "secure" customers. The "secure" customers will require more processor overhead, but not a lot.
So are your "secure" customers going to subsidize your free customers? Where and how will you make money?
The point of running a business is to make money, and you need to plan for that from the start. Simply saying that you think a competing service is making money isn't a plan for you. You need to plan for how you will make money in advance, or you will quickly run out of money and your business will fail.
I think it is one way to justify a charge, but may not be the deciding factor for a lot of users to convert. It may be a combination of: security, file space, speed, and the ability to share with other accounts. Some users pay just because they want to see a quality product flourish (I believe Evernote discovered this.).
There may be a market for health care professionals assuming you can meet their standards and keep an affordable rate. That will not be an easy task and will have issues from one country to the next.