This is usually due to Internet Explorer settings on the client machine.
They need to add the site  to their Local Intranet zone.

I've also ran across this problem. I added the site  to the trusted sites,
and also set the security level option to log-on with current
username/password (authenticated).

Note that that in Internet Explorer 6.0, Trusted Sites had 'logon with current
username and password' checked. IE7 does not have this checked for Trusted
Sites...

only Intranet has 'logon with current username  and password' by default in IE7.
You can change this, but it would require GPOs on a mass scale.

There are a couple of reasons this could be happening, Ben is right on about
making sure that login wirth current username  and password  is set in by GP
in your domain. Another reason this could be happening  is a change in the
application pool itself. Try resetting the app pool or reintering in the
service account and password your using. I'm assuming your using NTLM vs
Kerberos? If Kerberos there are a number of other things you could be
checking.

Kerberos
technet.microsoft.com/en-us/library/bb742516.aspx

Kerberos w SharePoint
blogs.msdn.com/.../configuring-kerberos-for-sh\
arepoint-2007-part-1-base-configuration-for-sharepoint.aspx

I have been studying this issue for some time, and I have found that it's a
complicated arena. I'd like to point out some hidden assumptions in the
solutions described so far:

1) All users  are on IE
2) All users are on Windows machines bound to the domain
3) All users are logged in on the desktop to AD accounts
4) All users want to use their desktop login for access to the SharePoint
site

These assumptions may be so acceptable they are invisible in corporate
settings, but at the university where I work, where students usually don't
fit assumptions 1-3, and staff sometimes don't fit assumption 4, these are
serious considerations.

Also, be aware that if NTLM is offered by the server to IE, and auto-logon
is NOT enabled (via browser settings described in previous posts), then the
user MUST specify the domain as well as username  (e.g., "AD\joeuser").
Otherwise the authentication fails and the user is prompted with IE's best
guess, "serverFQDN\username". Sometimes what users report as repeated
prompts results from not specifying the domain.

We have also found a server-side reason for multiple login prompts. I don't
know exactly how it happened, but at one point, we determined that part of
the SharePoint site  had NTLM and Basic authentication specified, and part
had only Basic. Since IE always prefers NTLM when it's offered by the
server, it was switching back and forth between authentication methods.
What made this especially maddening for users was that /layouts/images was
one of the affected folders, so they were bombarded with login requests to
see a single page. You can check for this condition using IIS Manager.