In my opinion, the example with "the" works much better than the example without, because you are talking about actually existing widespread awareness:
The widespread awareness that smoking kills has saved lives.
Back in the 1940s or so, when widespread awareness of that fact did not exist (that is, when awareness of that fact was not widespread), the version without "the" (and with "would save" rather than "has saved") would have worked well:
Widespread awareness that smoking kills would save lives.
Indeed, the "a" variant would also work, this indicating "a type of awareness that is widespread":
A widespread awareness that smoking kills would save lives.