Something that is a stable conjugate base doesn't 'need' the proton for stability. It is perfectly fine on its own. Its resistance to losing the hydrogen, therefore is low, meaning that the bond is weak and the proton can disassociate easily.
Now, to reverse it...if something is a strong base, that means that it is strongly attracted to the proton and is more stable in protonated form than without. That means that the conjugate acid is more strongly bonded to the proton and will resist losing it, making it a weaker acid.
Note that I reversed it by switching from 'stable' base to 'strong' base...these two terms do not mean the same thing. In fact, they are rather opposite. A stable base does not tend as strongly towards protonation, and can persist in an unprotonated state. A strong base tends to accept protons quickly, and has a higher affinity for protons than neutral water.