Update: The best answer is to use the new Database Monitor (for MS SQL Server)
[See Database Monitor information][1]
There are three approaches that work:
1. For databases installed on Windows, most (all?) of them will publish performance counters. So the Performance Monitor could watch the counters. Here's an [example counter list][1] list][2] from an MSDN blog.
2. The [Execute Script monitor][2] monitor][3] can periodically connect to a database and check something. The linked page has an example of how to do that.
3. Create a web page that connects to the database and then returns something like "OK" or "Problem". The [Web Page monitor][3] monitor][4] could then check that page.
[1]: https://www.poweradmin.com/help/pa-server-monitor-6-2/monitor-database-monitor.aspx
[2]: http://blogs.msdn.com/b/jimmymay/archive/2008/10/15/perfmon-objects-counters-thresholds-utilities-for-sql-server.aspx
[2]: [3]: http://www.poweradmin.com/help/SM_4_1/Monitor_Execute_Script.aspx
[3]: [4]: http://www.poweradmin.com/help/SM_4_1/Monitor_Web_Page.aspx