The other answer on this thread is excellent, but I just wanted to commend you on your correct reasoning in your initial post although it didn't directly lead you to the answer you wanted. You said that the more basic one would be the one where the conjugate acid (the protonated compound) is more stable. This is, of course, correct. Unfortunately, it is a harder route to see the difference when you think about it that way versus just comparing the bases.
To follow your initial line of reasoning, if you protonate the one with the double bond, it no longer has conjugation between the double bond and the nitrogen lone pair. So, the protonated double-bonded compound takes an energy penalty for protonation, while the all-single-bonds compound does not. That's why the protonated form of the double bonded thing is less stable relative to the unprotonated form. Make sense?