I don't know why Keurig is so popular when I read there are so many people having various problems. I too have a mini-plus one cup unit that leaks all over the counter from the bottom area. I have tried a couple of fix its for this.
Around the top puncture needle is a rubber gasket seal that can foul up with coffee grounds that prevent it from sealing tightly against the coffee pack. Remove the gasket by pulling gently down on it and clean it. Don't drop it down the sink. Reinstall the gasket and give the maker another try.
The second fix I did was to clean the upper needle by using a small paperclip. Open the paperclip on one wire and gently press it into the three needle openings to clean out any old coffee grounds. Gently doing this will prevent damage to the inside attached water tube. This seems to eliminate the water leak for a period of time. I'm not sure how exactly this resolves the leaking at the bottom but it does. My understanding is the discharge tube backs up because of these conditions and runs out an overflow tube located near the bottom of the unit.
Both these fixes may have to be repeated frequently if the leaking reoccurs.