You question intrigued me quite a bit, so I decided to try to solve this problem. Here is what I found and I would be grateful if anyone can help me improve my answer!
The first thing I knew was around 2007-2008 was about the time HD started getting popular in the United States. I used this information to track down a couple of forums on the AnimeNewsNetwork Here and Here. These two threads were very useful as they gave the names of "Samurai 7" and "Ghost in the Shell: Stand Alone Complex" (aka "GITS SAC" as used in the thread).
I pulled up the Wikipedia page for Samurai 7 and one part of it reads:
The series premiered across Japan on the anime satellite television network, Animax, as an exclusive high definition CS-PPV broadcast
Now I am not sure if you would like to consider it as aired in HD as it was "Pay Per View" (aka "PPV"), but it was created in HD based on this page about the series:
Based on Akira Kurosawa's classic movie, The Seven Samurai. Samurai 7 is being produced in High Definition at a price of $300 000 per episode, more than twice what an average non-hi-def series costs.
Now the reason I also included "Ghost in the Shell: Stand Alone Complex" is because in the forums many of the people were also saying that this one was also HD. I was able to find out that it was also aired by Animax, but cannot confirm if it was aired for HD.
If GITS SAC was not aired in HD then it would make Samurai 7 the first anime aired in HD (or at least as far as I can find).
Now I am very weary about my research, as I think there may have been other anime's aired in HD before these two, and my main concern comes from this article written in 1998. And the way the users in the threads were talking about HD as it was not something new, also makes me wonder if there have been anime's before these two that were aired in HD.
I will continue to do more research and improve my answer as I go.
I hope this answer can help you.
I was able to find this question, where "Toshinou Kyouko" pointed to this thread. In the thread "Zalis" states:
First 16:9 TV anime: Betterman (Spring 1999), Vandread (Fall 2000)
I am currently trying to find a source that can verify if these shows were created in HD 16:9 aspect ratio. (Thanks to "Ross Ridge" for pointing this out.)