Try stripping your code down to a sample that tests the binary search:

use strict;
use warnings;

my @allwords = sort qw(array will be the sorted wordlist and target);

sub binary_search {

    my ($array, $target) = @_;
    #set arguments for future use : $array will be the sorted wordlist
+ and $target, the word we will be looking for.

    my ($low, $high) = (0, @$array - 1);
    #Declare high and low indexes. Low index = 0 and high index = last
+ index of the array.

    while ($low < $high)
    {    # If high index is higher than the low index, keep the window
+ open.
        my $cur =
            int ($low + $high) / 2
            ; #Declare a middle, which is the total of high index and 
+low index /2.
        if ($array->[$cur] lt $target) {
            $low = $cur + 1;    #If the target is too small, try lower
+.
        } else {
            $high = $cur;       #Else, try higher.
        }
    }
}

for my $word (qw (search wordlist)) {
    my $index = binary_search (\@allwords, $word);

    next if $index >= @allwords;
    
    if ($allwords[$index] eq $word) {
        print "Found $word at $index\n";
    } else {
        print "Couldn't find $word\n";
    }
}
[download]

Prints:

Couldn't find search
Couldn't find wordlist
[download]

which seems to indicate that your search is failing.

For any non-trivial project you would take the test code and wrap it up in something like Test::More as a unit test so that you have a convenient way to test all the edge cases (zero, one or two words in the list for example) whenever you alter the code.