In my view, it will depend on your likely customer base. If you could get into a situation where arch-rivals are both using your system, then you would be better off with separate databases. It also depends on how multiple databases get implemented by your DBMS. If each database has a separate copy of the infrastructure, then that suggests a single database (or a change of DBMS). If multiple databases can be served by a single copy of the infrastructure, then I'd go for separate databases.
Think of database backup. Customer A says "Please send me a copy of my data". Much, much easier in a separate database setup than if a single database is shared. Think of removing a customer; again, much easier with separate databases.
(The 'infrastructure' part is mealy-mouthed because there are major differences between different DBMS about what constitutes a 'database' versus a 'server instance', for example. Add: The question is tagged 'mysql', so maybe those thoughts aren't completely relevant.)
One more issue - with multiple customers in a single database, every SQL query is going to need to ensure that the data for the correct customer is chosen. That means that the SQL is going to be harder to write, and read, and the DBMS is going to have to work harder on processing the data, and indexes will be bigger, and ... I really would go with a separate database per customer for many purposes.
Clearly, StackOverflow (as an example) does not have a separate database per user; we all use the same database. But if you were running accounting systems for different companies, I don't think it would be acceptable (to the companies, and possibly not to the legal people) to share databases.