There are a few simple approaches that work for easily monitoring databases.
In addition, other database servers like MySQL can also be monitored via SNMP. Other databases likely have similar functionality.
The Execute Script monitor can periodically connect to a database and perform a check. The first example script shows how to do that. This method even lets you easily check on a database value (table size for example).
Create a web page that connects to the database and then returns something like "OK" or "Problem". The Web Page monitor could then check that page and alert if it didn't see "OK". That web page could of course perform any sorts of checks that you need.