This is a prompt from Visual Studio and not NuGet. Information entered here is not saved with NuGet. Although it should be cached for the current Visual Studio session.
NuGet stores user names and passwords in its NuGet.Config file. You can save usernames and passwords for protected feeds by adding them to the NuGet.Config file using the nuget.exe command line.
NuGet.exe Sources Update -Name <feedName> -Source <pathToPackageSource> -UserName xxx -Password <secret>
If this is a prompt for a proxy then you can either set it using nuget:
NuGet.exe config -Set HTTP_PROXY=http://127.0.0.1 -Set HTTP_PROXY.USER=domain\user
Or use an environment variable http_proxy with the value specified in the format http://[username]:[password]@proxy.com