I have basic authentatication working with REST API using curl:
curl -X POST -H \'Accept: application/json\' -u user:password http://localhost/test/
I know this is an old thread, but for those who might stumble across this the invoke-restmethod is a much better, and simpler, vehicle for making API calls with PowerShell.
Build a parameter list as a hash table:
$params = @{uri = 'https:/api.trello.com/1/TheRestOfYourURIpath';
Method = 'Get'; #(or POST, or whatever)
Headers = @{Authorization = 'Basic ' + [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes("$($acctname):$($password)"));
} #end headers hash table
} #end $params hash table
$var = invoke-restmethod @params
Your parameter hash table may differ slightly.
I actually haven't gotten this to work with Trello, but I have with GitHub, Serena Business Manager and Jira.