((17((5[3-9])|([6-9]\d)))|((18|19)\d\d)|([2-9]\d\d\d))[-/.](0[1-9]|1[012])[- /.](0[1-9]|[12][0-9]|3[01])   
                                                        ^ unescaped / terminates the RE if you use it in /((17...)/
but even if you encapsulate it in a variable there's still problems:

#!/usr/bin/perl

@sampdata = qw (
894/7/14
1752/8/12
1753/12/24
1957/8/30
3998/4/22
9999/3/15
10000/1/1
);

$re = "((17((5[3-9])|([6-9]\d)))|((18|19)\d\d)|([2-9]\d\d\d))[-/.](0[1-9]|1[012])[- /.](0[1-9]|
[12][0-9]|3[01])";

while (<@sampdata>) {
        print;
        print ( /$re/ ? " is " : " is not " );
        print " in range 1753 to 9999\n";
}

__END__



894/7/14 is not  in range 1753 to 9999
1752/8/12 is not  in range 1753 to 9999
1753/12/24 is  in range 1753 to 9999
1957/8/30 is not  in range 1753 to 9999
3998/4/22 is not  in range 1753 to 9999
9999/3/15 is not  in range 1753 to 9999
10000/1/1 is not  in range 1753 to 9999

So a change of 
    ((17((5[3-9])|([6-9]\d)))
to 
    ((17((5[3-9])|(1([6-9]\d))))
Seems to be in order.

Re: the two instances of [-/.], was the second one supposed to include a space?