MySQL looks at usernames not as 'username' but as 'username@ip_addresss'. Off the command line this generally ends up as 'username@localhost' which is the default allow when you do a grant. In the mysql database user table the Host field can be amongst other things 'localhost' - allow local conns for that user or '%' which as expected allows connections from an ip that matches '%' ie all. Quickfix:

DBI->connect("DBI:mysql:database=DATABASE_NAME",
                         "USER_HERE@localhost", 
                         "USER_HERE_PW",

If you actually mean really remote access you need this syntax:

$port = 3306;
$dsn = "DBI:mysql:database=$database;host=$hostname;port=$port";
$dbh = DBI->connect($dsn, $user, $password);

# more details on this (you could update the Host field from 'localhos
+t' 
# to '%' and FLUSH PRIVILEGES to get it to take effect.

mysql> use mysql

Database changed
mysql> show tables;
+-----------------+
| Tables_in_mysql |
+-----------------+
| columns_priv    |
| db              |
| func            |
| host            |
| tables_priv     |
| user            |
+-----------------+
6 rows in set (0.00 sec)

mysql> select * from user\G
*************************** 1. row ***************************
                 Host: localhost
                 User: root
             Password: bipitybopityboo
          Select_priv: Y
          Insert_priv: Y
          Update_priv: Y
          Delete_priv: Y
          [snip]
        max_questions: 0
          max_updates: 0
      max_connections: 0
*************************** 2. row ***************************
                 Host: %
                 User: remote_god
             Password: TheCanBeOnlyOne
          Select_priv: HeavenOrHell
          Insert_priv: Adam&Eve
          Update_priv: Birth
          Delete_priv: Death
          [snip]
        max_questions: Eternal
          max_updates: Last try failed
      max_connections: ?
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