$user = "USERNAME"
$pass = "PASSWORD"
$pair = "${user}:${pass}"

#Encode the string to the RFC2045-MIME variant of Base64, except not limited to 76 char/line.
$bytes = [System.Text.Encoding]::ASCII.GetBytes($pair)
$base64 = [System.Convert]::ToBase64String($bytes)

#Create the Auth value as the method, a space, and then the encoded pair Method Base64String
$basicAuthValue = "Basic $base64"

#Create the header Authorization: Basic QWxhZGRpbjpvcGVuIHNlc2FtZQ==
$headers = @{ Authorization = $basicAuthValue }

#Invoke the web-request
Invoke-WebRequest -uri $uri -Headers $headers