I was able to download the data just fine using the WIST website (https://wist.echo.nasa.gov/api/). The interface is complex and confusing, so the tutorial link at the top of that page is somewhat helpful.
You must create an account before anything else, or it won't work correctly. Once you've done that, logging in is also confusing -- you need to log in first, click the Login button, the page refreshes and then you can complete the rest of the search form (in other words, you can't log in and search at the same time even though they're on the same page).
For the ASTER DEM data, choose ASTER from the Land column, then choose ASTER Global Digital Elevation Model from the list below the heading "Choose Keywords for One or More Categories". To order, they ask for your physical address and it looks as though they're going to send you a CD, but a few minutes after submitting my order, I got an email with an FTP download link. The data I looked at was very good and had no holes or weird artifacts.
One more thing: the interface only works with some browsers, such as IE7, but not Google Chrome.