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Digging around a bit, it appears WebKit itself doesn't handle authentication at all.
 Instead, it would appear on Android that the package org.apache.http.auth does this.

It's definitely not passing through the information.  Here's a snippet from my Apache
log:
---
192.168.10.199 - - [20/Nov/2008:20:14:37 -0600] "GET /file.txt HTTP/1.1" 401 401 "-"
"AndroidDownloadManager"

is there any news about custom map types ?

Any update on this issue?

Apparently there isn't anything else to say concerning this is there.

Soooo, yeah, just confirmed this on several devices. It would be really nice to be
able to access my server from my phone. That is kind of why I set it up.

Maybe it would work on another browser besides default? Anyone know?

So I devised a workaround for this issue if anyone is interested. However, you have
to allow free access from any browser that broadcasts itself as the
AndroidDownloadManager to the folders you want your phone to be able to download.
Here is what you need to enter in the httpd configuration file in apache:

SetEnvIf User-Agent ^AndroidDownloadManager let_me_in
<Directory "FolderWhereYourStuffIs">
    Order deny,allow
    Deny from all
    Allow from env=let_me_in
    require valid-user
    Satisfy any

I did this for every individual folder, because there are some folders in my root
directory that I do not want to put online and if you put this in the root directory
then you have to share everything because of the satisfy any directive. This
directive will be applied to every sub-folder, and there is not way to stop it.
(Maybe a limit phrase, but I am not advanced enough to figure that out).

@hunterpritchett:
I tried it with the dolphin browser too and it doesn't work (without the modification
suggested later). My impression is that there is one download manager which handles
file downloads and this is used by dolphin too. When I clicked on the file, dolphin
showed up the download history used by the default browser.

I have posted this problem some time ago also on the google nexus one forums:
http://www.google.com/support/forum/p/android/thread?tid=2a6aeb6741c24a7a&hl=en
where I also add that the credentials are also not passed to any other program. So
even if I download a playlist from the password protected site and I try to play it,
it does not get played as the authentication of the browser is not passed on to the
media player.

It's certainly an issue and I would love to have a solution too, if anyone is
listening :). Apart from a few bugs, very happy with the N1.

Opera seems to be able to download files just fine.  It uses it's own built-in
downloader.  I have the same Apache setup and the same difficulties with the Webkit
browser, but Opera has no difficulties.

Awesome guess my workaround is only necessary for the other browsers then. I have the
opera browser anyway, so that's great.

This defect has not been fixed in Froyo. Please fix this bug, as it prevents me from playing music from my server.

I've run into the same problem with my Nexus One running android 2.2.  After basic http authentication, I can download html files with the android browser but not mp3 files.  My iPod Touch running mobile safari (also webkit based) downloads the mp3s without difficulty.  I hope Google will fix this soon.   I posted issue 36919571 before I found this issue, #1353, had been submitted.