Use getOpenFile to show the files to the user.

#! /usr/bin/perl
use strict;
use warnings;

use Tk;
use Path::Tiny qw{ path };

my $safe_path = qr{^/safe/path};

sub select_and_run {
    my ($mw, $text) = @_;
    my $scriptfile = $mw->getOpenFile(-filetypes => [[Perl => '*.pl']]
+);
    my $result;
    if ($scriptfile =~ $safe_path && -x $scriptfile) {
        my $output = qx{ $scriptfile };
        my $exit_code = $?;
        path("$scriptfile.output")->spew($output);
        $result = "Exit code: $exit_code";
    } else {
        $result = 'Cannot run';
    }
    $text->insert(end => "$result\n");
}

my $mw = 'MainWindow'->new(-title => 'Script Runner');

my $f = $mw->Frame->pack;
my $text;
my $button = $f->Button(-text    => 'Run Script',
                        -command => sub { select_and_run($mw, $text) }
+)->pack;
$text = $f->Text->pack;
my $q = $f->Button(-text    => 'Quit',
                   -command => sub { Tk::exit })->pack;

Tk::MainLoop();
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Update: Minor code fixes.