Player A plays "Take Initiative".

Player B plays "Negation" - Does the "Take Initiative" play allow Player B to utilize his Agent Blaise to trigger Adapt?

After (re)reading the FAQ for RtH: "Negation cancels all effects of the affected card, etc"

In other words, if the command card play is cancelled by Negation, does Agent Blaise get to use Adapt?

I say you react with 'Adept' before Negation is played. So yes, you can do that.

Adept says when you play the card, not when it is resolved.

I would also say that Adept isn't an effect of the card, it's an ability of the figure. Take Initiative is still being played, which is the trigger for both Adept and Negation.