Setting up SQL Server to handle queries from external domains
It depends on what you mean by a proper user name and password. 

Windows Authentication

You will need to setup the Domain trust relationship between each domain. Your domain hosting the database server will need to trust the domain holding the web application. This should allow the domain accounts access the web application can connect to the database server, without having to use a separate account.

SQL Login

If you do not want to deal with the domain relationship you can simple configure SQL Server in mixed mode. You can find those instructions here. You would just create a SQL Login and grant the required access to the database(s) for the web application.