I think it sounds old fashioned.
However, I think the interpretation would depend a lot on the delivery, speaker, and audience. For example, someone could be using it ironically, and if the the audience were able to notice the irony, then it might sound ironic rather than old fashioned1.
Why the phrase old fashioned?
I'm not Noam Chomsky or a linguist, but I think that the reason that it sounds old fashioned is because it's wordy, and overly polite and submissive sounding. I think that the current trend is to speak concisely and to be respectful rather than submissive or polite.
I also think that the phrase itself is a little ridiculous and empty, and this might contribute to the interpretation. Afterall, if the answer is "yes" 2 then of course you don't mind. And anyway, why would you ever mind your own actions? You would sound a pretty crazy if you said "I really mind if I eat this, but I'm going to eat it anyway" (but nobody would be evaluating whether or not you sound old fashioned).
I didn't notice other answers actually diving into why they feel this phrase sounds old fashioned, only giving statistics or examples. Answering "why" is a harder problem, but I think it's part of the answer. I would definitely defer to someone who actually studies language, and I would be interested in their answer. Although I don't care about this phrase in particular, I think the evolution of language is interesting and important.
- Note that that the phrase is still an old fashioned phrase, it's just that you're using it ironically. To borrow the Homer Simpson example, Homer's saying "Well I don't mind if I do" in an effort to sound silly, and part of the reason that he sounds silly is because he's using an old fashioned phrase, and part of the reason is because he's pretending to sound polite, which you might also say is also old fashioned (just take a sample of comments anywhere on any StackExchange site to observe this).
- By "yes" I mean any variation of agreement which can be reduced to the affirmative. "Yes" is used here for brevity and clarity.