No less irresponsible, either. I only use such features if they in turn are password-protected and encrypt the stored data with that passphrase. And mostly I don't even use them; I use Lastpass instead.
Only you play on that computer; but you won't be the only one who checks that option off. Others who do so won't be the only one using their computer, and when their roommate/sister/boyfriend/neighbor/fratmate uses their account, they'll be contributing to the length of the support queue.
Further, since that data will not be in any meaningful way protected, it will be accessible to malware. Gold sellers employ programmers to create malware. The support floods when they have a big strike make the above look like small beer.
So? Thats no reason for it to not be an option. Thats the risk that is taken by everyone who uses a "remember this password" function. But its still an option, and Ive not read of there being any major problems with it in any other software.