Both tools (MySQL and MS-Access) can be secure with a competent admin. There are lots more competent admins for MS-Access than there are for MySQL. MySQL is somewhat easier to learn to secure, especially since there are lots fewer well trained attackers.
In the long run, you are between the devil and the deep blue sea. Neither is inherantly secure, both can be secured if you do them right. Other products such as MS-SQL, DB2, and Oracle can be configured more securely, more easily.
-PatP