Without much work this is not possible if the server and client computers aren't in the same domain.
In a domain environment, authenticating for Sharepoint access the process in the background works roughly like this:
- First you authenticate to domain controller
- Domain controller assigns you a ticket
- Web server checks your ticket
In non domain environment it is simply not trivial to configure such automatic authentication on a Serverside.
You can configure automatic authentication on the client side by storing password in the browser or by configuring IE to authenticate with current user credentials.
When navigating to Sharepoint site new dialog box opens up, asking for credentials. After you supply the credentials, a "dialog-line" appears at the top of Firefox browsing area, asking if user wants to store password. By selecting OK, credentials get stored. Next time visiting a site user simply confirms autocompleted form with Username/password.
The approach mentioned for firefox didn't work for me. IE does offer to remember password, but even if you check this option, credentials don't get stored. (Maybe someone has solution for that).
The following is what does work: You add the site to Local Intranet Zone (Tools -> Internet options -> Security -> Local Intranet -> Sites -> Advanced). By default Security level is set up in the way that IE should attempt to authenticate you automatically using current user credentials. You can check that by clicking Custom level, then search for user authentication section. If either Automatic logon only in intranet zone or Automatic logon with current user name and password is selected, this shoud work.
Of course instead using Intranet zone you can add site in Trusted zone or somewhere else. Just make sure that Automatic logon with current user name and password options is selected.