I have this problem too, but only in some oddball sites, such as job application form website/tools or similar forms where it would be much, much easier to copy and paste from another document. Your situation might be something to do with the Safari preferences file being corrupted, or the cache or something buggy like that.
I'd like to know why, so at least I would know why I'm frustrated. Why is it? What is it about these sites' code that makes something so basic and simple-seeming as copy and paste not work? Some guru here must know. ? :-) I'd appreciate just knowing the technical reasons why even if there's no work-around.
In one case, the vendor of a grad-school application tool on a university website - the tool seems to be owned by Hobsons - recommended I (basically) troubleshoot it for them by using VirtualBox (like Parallels but free) and IE for Windows; however, they don't seem to realize I will have to buy a copy of Windows OS in order to run IE through VirtualBox.
Regarding your situation, here is the advice I've found so far.
1) try using "paste and match style" instead of just paste (I don't know why, but maybe it strips some codes)
2) try pasting to a true text document (as in TextEdit) first, then copy paste that to the text box
3) Close and restart Safari, just to be sure it's clean when the situation happens.
4) Close Safari, then try trashing the Safari plist/preferences files under your user account/library. See Carolyn's tip here https://discussions.apple.com/docs/DOC-3110
5) Empty the cache in Safari, restart Safari, and try again.
See if you can do it using Chrome or Firefox instead of Safari; it might be easier anyway.