No, this is how I understand it ;

Support for pre-4.1 password hashes was removed in MySQL 5.7.5
Strawberry Perl 5.24.4.1 is compiled with mysql-5.7.16 so will not work with any user account that has a short (16 byte) password hash like your current one.
It's the version of mysql library that's important not the version of the DBD-mysql module.

These versions should work with 16-byte hash (or 41-byte hash)

Perl 5.20.3   (2016-03-08) comes with DBD-mysql 4.033 and mysql-5.1.44 so post 4.1 and pre 5.7.5
Perl 5.24.0.1 (2016-05-11) comes with DBD-mysql 4.033 and mysql-5.1.44 so post 4.1 and pre 5.7.5

This version should work only with 41-byte hash.

Perl 5.24.1.1 (2017-01-16) comes with DBD-mysql 4.041 and mysql-5.7.16 so post 5.7.5.

1. For existing user (with 16-byte hash) generate a new 41-byte hash using 
     SET PASSWORD FOR 'some_user'@'some_host' = PASSWORD('password');

   If you do that any existing pre-4.1 client will not be able to use that user account.
   
   For info the first production version of mysql-4.1.7 was released
   in October 2004.
  
   note. PASSWORD_OLD('password') generates a 16-byte hash.
  
or 

2  Create a new user account (which will be by default a 41-byte hash) and use that.    
   Any pre-4.1 clients could continue to use the old account.