I'm assuming that by multiple customers you're not just storing customer information, you're hosting databases for an application for the customers, like CRM systems.
If so, then I would absolutely not store everything in the same database.
Reasons:
- Backup, when one customer calls and says that he needs to restore a backup because an intern managed to clean out the production database and not the test database, you do not want to have to deal with all the other customers at the same time
- Security, even with a bug in the application it won't be able to reach data for other customers. Also, consider if one customer is a bit too relaxed in their own security considerations and leaks passwords or whatnot to the system, if hackers discovers a way into that customers database, consider the fallout if that also includes all other customers you're hosting for.
- Politics, some customers will not allow mixing their data with other customers even if you can 100% guarantee that access to their data won't be (accidentally) given to other customers
So bottom line: separate databases.