No, the base is not to be full of water, it should be dry. I would suspect a leaking washer for the aerator is causing water to drip/spray back inside the plastic cover under the spout. I would unscrew the old one, making sure the washer also comes out, and replace it with a new one, and washer.

Do you have water in the base? Water should be running thru tubing that is within the base.

If the aerator isn't the problem and the drip is occasional, then it's probably just from wet hands using the lift rod. The area the rod goes thru is not water tight.

randy,
When you speak of the washer, do you mean the 0 ring?
That came with the new cartridges.

annz,
I was thinking the base might have water in it for it to leak down through the lift rod hole?
The leak is after I use the faucet each time. Then it stops.
No sign of water around the faucet on the sink.

No, the washer functions to form a water tight seal for the aerator that is screwed in at the water spout. It's not part of the cartridge installations.

Funny thing------

This morning, I shut both valves off on the water supply for each faucet.
I took the cartridges out and pressed a rag in the opening to soak up the water that was left.
Then I poured some water back in and it dripped from where I explained above, from the lift rod

annz, then it's probably just from wet hands using the lift rod., For sure not. I am not using the lift rod when that happens nor have that much water on my hands.

If I get brave enough to see if I can take the faucet off, I shall investigate under the gasket etc. on the sink.

randy,
I would suspect a leaking washer for the aerator is causing water to drip/spray back inside the plastic cover under the spout. I would unscrew the old one, making sure the washer also comes out, and replace it with a new one, and washer.
By golly, I think you ARE right?
I was playing around with the aerator to find it not in the best shape and have to see about repairing it. (Labor Day closed)