If one of the tails is leaking it wouldn't only happen when the tap is on.
Have seen similar symptoms caused by the water running down the underside of the spout and getting through the tap hole of the basin/sink.
Your last comment about it only happening when hot is turned on - if your hot water flow is lower than your cold then the water would be more inclined to run back down under the spout.
Is water pooling at the base of the tap when it is operated?