And to clarify, I don't think any of this solves the OP's actual problem. That would certainly require more from the script. But I just remember banging my head with -contains in the past and wanted to save some poor soul some time.

With all due to respect, I don't think that script would work.

help about_comparison_operators states -contains "Returns true when reference value contained in a collection". So it wouldn't apply here. I remember this tripping me up in the past. -match is what you're looking for. Specifically I'd use -imatch for a case-insensitive match.

And if I may, here are couple other things I'd change.

I'd join these two lines together:

$folders = Get-ChildItem -Path $baseFolder -Directory |
    Where-Object -Property Name -imatch $matchString

Rather than manually concatenating the destination path, I'd use the Join-Path cmdlet. And since there are more than two parameters I'd use a splat.

foreach ($folder in $folders) {
    $copyParams = @{
        Path        = $folder.FullName
        Destination = Join-Path -Path $destFolder -ChildPath $folder.Name
        Recurse     = $true
    }
    Copy-item @copyParams
}

Hope this helps!