A small note to start with:

#------Check options-----
# Check if host is defined and alive
if (!defined $host){
    print "\nNo host specified\n\n";
    showUsage();
} else {
    print "\nChecking if $host is alive.. \n";
    $ping = Net::Ping->new();
    if ($ping->ping($host)){
        print "$host is alive\n\n";
    }else {
        print "$host is unreachable\n\n";
        exit;
    }    
    $ping->close();
}
[download]

That reads awful for several reasons

• Reverted if branches if (not expr) {} else {} => if (expr) {} else {}
• showUsage () does an exit which is unclear from the code, as showUsage is at the bottom.

And you should make clear difference between wanted output (STDOUT) and error/diagnostics (STDERR).

I'd put showUsage () at the top of the script, which also functions as immediate documentation for someone starting to read the script, making that if/else easier, like:

sub showUsage
{
    my $err = shift and select STDERR;
    print "\n------------------------------------------\n";
    print "-- Usage for the database backup script --\n";
    print "------------------------------------------\n";
    print "\n--host \n";
    print "      Specify the host IP\n\n";
    print "--engine \n";
    print "      Specify the database engine. This could be postgres o
+r mysql. \n\n";
    print "--password \n";
    print "      Give the password needed to connect with the database
+ server\n\n";
    print "--user \n";
    print "      Specify user to connect to the database server.\n";
    print "      The user must have permissions to connect to the data
+base(s)\n\n";
    print "--port \n";
    print "      If another port than the standard must be used to con
+nect you could \n";
    print "      specify this with the --port option\n\n";
    exit $err;
    }

GetOptions (
    "help|?"     => sub { showUsage (0); },
    "host=s"     => \$host,
    "engine=s"   => \$engine,
    "password=s" => \$password,
    "user=s"     => \$user,
    "port=s"     => \$port,
    "all"        => \$all,
    ) or showUsage (1);
# no need for a $help option here

#------Check options-----
# Check if host is defined and alive
unless (defined $host) {
    warn "\nNo host specified\n\n";
    showUsage ();
    }
# That exits, so no need for an else

print "\nChecking if $host is alive.. \n";
Net::Ping->new ()->ping ($host)
    or die "$host is unreachable\n";

print "$host is alive\n\n";
[download]

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