There are only two options.

1. If you use Integrated Windows authentication  and add the address of the
SharePoint site  to the Local Intranet Zone it will pass the user's operating
system credentials in the background. If the computer is a member of the domain
they will be automatically logged in and will not be given the login  prompt.
They'll still get the login when they try to do some things like edit a
document, at which time  they must supply "domain\username" and "password".
There is no way to get around this if you are using NTLM and Integrated Windows.

2. You can switch to a different authentication system like Forms Based
Authentication where the username  can be set to be in the format of
username@domain.

A third option is to use Basic auth over SSL. Basic auth lets you define
a default domain  in IIS.

If I enable Basic authentication  but without SSL, is it a very bad
practice? The sharepoint  site is an intranet site  only.

Any time  you transmit clear credentials across a wire it is a bad idea. Most
cracking is actually done inside the network.

I am wondering why Windows does not allow encrypted authentication  +
default domain

It does - that's Basic over SSL......

Do you think if I do the following steps, the password  still transmit in
clear text?

"

I just did some test and found out on the IIS manager if I

1. Check "Basic authentication"

2. Set default domain

3. UnCheck "Basic Authentication"

4. Click OK

The authentication  will work for both "domain\user" and "user"

"

Passwords are never in clear text, but Basic uses a really simple
mechanism that isn't encryption. Anytime you use Basic, it's as good as
clear unless you also use SSL.

What you describe is using either NTLM or Kerberos (no way to tell
without knowing more about your server settings). Neither of these
actually send your password  as part of the request/response, and are
therefore more secure than Basic, but there are still exploits possible
if you aren't using SSL.

I just did some test and found out on the IIS manager if I

1. Check "Basic authentication"

2. Set default domain

3. UnCheck "Basic Authentication"

4. Click OK

The authentication  will work for both "domain\user" and "user"

You might try doing some targeted google searches using the term "default
domain". Of course this won't solve your problem if the users  reside in
more than one domain...