It looks like you never assigned $line to anything, and are searching its contents, which are undefined. You may be getting confused about how Perl's default $_ is used in pattern matching. The program would do what you want if you left off the variable my $line and changed the if statement to read:

... since $_ has the line data in it, not $line. Or conversely you could put the line data into the variable like this...

while( my $line = <FH> ) {
    if ( $line =~ m/A-Za-z/ ) {
        ....
    }
}
close FH
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The
    while (<FH>) { ... }
construct reads by default to the special scalar variable $_ (dollar-underscore; see perlvar), but you are matching against the variable $line, which is never defined.

Use of strictures and warnings is very wise, but not in the 'shebang' line, where they are ineffective: use them separately in the body of the script.

Try something like the following (untested (Update: changed some variable names to be self-documenting)):

#!usr/bin/perl

use warnings;
use strict;

my $filename = $ARGV[0] or die "no file name given";

open my $fh, '<', $filename or die "opening '$filename': $!";

my $previous_line = '';
while (defined(my $current_line = <$fh>)) {
    chomp $current_line;
    if ($current_line =~ m{ \A [ACGTacgt]+ \z }xms) {
        print "$previous_line\n";
        print "$current_line\n";
        }
    $previous_line = $current_line;
    }
    
close $fh or die "closing '$filename': $!";
[download]

This will print every line that contains only base-pair letters, along with the line before it in the file. When this works, modify the
    m{ \A [ACGTacgt]+ \z }xms
regex to match whatever a 'motif' is.

Update: Since it looks like you're working with a FASTA-formatted file, check out the FASTA-related modules in CPAN; I'm not a bio-guy, so I can't advise which might be best for you.

See also Perl and Bioinformatics