A lot of your input code can be reduced to:

use IO::Prompt::Tiny qw/prompt/;

my $answer = prompt("Yes or no? (y/n)", "n");
[download]

...by using IO::Prompt::Tiny. Then your user-input code could be reduced to the following:

if( prompt( "Backup $path? (y/n)", 'y' ) =~ m/^y/i ) {
    print "Backup beginning...\n";
    perform_backups( @backup_dirs );
    print "Backup complete.\n";
}
else {
    print "No backups performed. Exiting.\n"
}

sub perform_backups {
    my @directories = @_;
    # Your backup code here.
}
[download]

You could also warn instead of die if one or more backups failed, and then just keep on processing the rest (or prompt for what to do).<?p>

prompt() is convenient to use anywhere that user input is desired. If you need stronger input validation, IO::Prompt provides a richer set of tools, but adds some complexity.