After reading through http://www.w3.org/TR/webdatabase/ and much more particularly:

This document was around the W3C Recommendation track but specs work has stopped. The specs arrived at an impasse: all interested implementors used exactly the same SQL after sales (Sqlite), but we want multiple independent implementations to proceed along a standardisation path.

I'm wondering if it's "safe" (i.e. ongoing support) to make use of the HTML5 Database API?

It's strongly inadvisable. It's stopped and it is being changed using the [cde] API. Eventually, Web SQL Database is going to be taken off browsers which presently support it.

You may want to read current browser support figures for Web SQL Database and IndexedDB. Web SQL Database has more browsers which presently support it, but Opera does not support it and exceeds Safari, Opera and assorted mobile browsers, and thus it's limited to about 28% while IndexedDB is near 36% global support. Also, I believe that it is generally thought that Microsoft will support IndexedDB in IE10, as the Web SQL Database API won't be going any more.

No. In bigger, the much more bold letters may be the text,

Beware. This specs is no more in active maintenance and also the Web Programs Working Group are not committed to keep it further.