How many databases do I need?

Multiple installs can share a single database, as long as each install has a unique table prefix defined in wp-config.php before going through the install process:

/**
 * WordPress Database Table prefix.
 *
 * You can have multiple installations in one database if you give each a unique
 * prefix. Only numbers, letters, and underscores please!
 */
$table_prefix  = 'client_1_';

I personally find it easier to just create a new database for each install, as my host allows an unlimited number of databases. I do still however make the table prefix unique for every install, as it could possibly protect against SQL injection attacks if a client installs a bad plugin with insecure code.