Unfortunately, the magazine's website doesn't have anything useful to say. So one would imagine that standard cover letter advice would apply here. (I got some of this information from the links at the end of this answer, but have condensed, and reworded significantly.)
- Your name, contact info, and a word count
- Follow their cover letter guidelines (where they have them - in this case there are none)
- The letter should be to a specific person, when you can do that (for The New Yorker, just "Fiction Editor, The New Yorker")
- Mention the magazine, but briefly. (So that it's clear your note isn't a form letter)
- Is this a simultaneous submission? Some magazines want to know that. Some don't allow it at all. (Check their guidelines.)
- Keep it short. Magazines get a lot of submissions, and a long cover letter is unlikely to be read.
Since the New Yorker doesn't really say what they need, I'd either contact them and ask for clarification or find out what other writers have done. Do they even want a cover letter? Maybe this info should just be in the email? Or maybe they just want an email address and nothing else? Ask, or put together a very brief letter. (A short cover letter is unlikely to be a problem.)
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