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1) Encode the credentials on the system where the script will be run:
 
This is widely used to access alternate credentials, of course in this case you wouldn't be using Anonymous, instead you would use some other account that has permisison to send, domain admin or whatnot.   Follow this procedure:
 
One time operation to create the encrypted password file:  
      Read-host -Prompt "enter password" -AsSecureString |
      ConvertFrom-SecureString | Out-File C:\AdminPassword.txt
 
In your script pull in the encrypted string from the file, convert it to a secure string, and cite it as an argument to the your new credential object.  
      $pass = Get-Content C:\AdminPassword.txt | ConvertTo-SecureString
      $creds = New-Object -TypeName System.Management.Automation.PSCredential -ArgumentList <UserName>, $pass
 
Execute the Send-MailMessage command using the -Credential parameter with $creds as it's argument.
Don't forget to fill in your own information for the above commands.   Also make sure you encode the file on the system and logged in under the account that will run the ultimate script.
 
2) You can create an account with very limited access and permission your ReceiveConnectors to allow it to send.  You can combine this approach with option 1 so you can use those credentials in scripts without the same security risks.
 
3) If you really want to send using the Anonymous account you'll run into a problem.  I appreciate the suggestion to use an empty credential object, but the cmdlet isn't expecting that.  Using Anonymous requires a credential object an Anonymous user but with an "empty" password.  You cannot pass a null or empty string as the password argument in the New-Object command I previously showed.  In fact that argument expects a SecureString object, what's worse is you cannot pass an empty or null string to ConverTo-SecureString so you can't prep your argument beforehand.  The question becomes, how can you get an empty string into a SecureString Object?  Well Read-Host -AsSecureString can do it, if you just hit enter at the prompt. But that's interactive and won't serve us if we want this script to run by itself .  Moreover, I assume if we attempted to take the output of Read-Host and put it in a file as we did in option 1 above we would again have issues with ConvertToSecure-String not liking the empty string.  The solution to this dilemma is kind of crafty:
 
   Use a any old string instead of a empty string:     
      $pass = ConvertTo-SecureString "whatever" -asplaintext -force
      $creds = New-Object -TypeName System.Management.Automation.PSCredential -ArgumentList "NT AUTHORITY\ANONYMOUS LOGON", $pass
 
   Execute the Send-MailMessage command using the -Credential parameter with $creds as it's argument.  
Hard to believe but this actually works.  The obvious benefit of this approach is you don't need to store or create anything, and you can use Anonymous instead of a real account with potentially greater and/or more dangerous access.  Even though options 1 & 2 are pretty secure many administrators or even security officers may have a problem with stored credentials.  There's certainly potential for abuse, but option number 3 seems to work completely with no exposure other than what's been explicitly granted Anonymous.