Do you even trust the site to actually do what it says? After all, a web application is just [someone else's] code running on your system.
Even if it's secure against XSS, HTTPS is intact, etc, you are ultimately putting presumably sensitive data into code someone else wrote and then allowing it to run on your machine with explicit network access.
The crypto implementation may be flawed, or the developer may even send your private key up or other data that you don't expect it to be sending (whether it is out of lack of understanding, accident or malice). It would be nearly impossible for you to spot if it's just sent as part of the big blob of encrypted data you expect it to be sending. The blob being bigger than you expect might be a clue, but the only way to know for sure is to audit the code, and if the code if obfuscated/minimized that could be quite the task in itself.
This is really not a problem of browser crypto at all; this is a problem of trust of the code you're executing and using.