The problem is that over the years, the spot where the bearing seal runs has worn down. You can feel a groove in this part of the axle with your finger.

But how do you repair this? Replace the axle shaft??

Well I stumbled on a cheaper solution. Bearing Buddies, the same people that make the pressurized grease cups also makes a seal retrofitting product that corrects leaking seals., its call 'Spindle Seals'
http://www.bearingbuddy.com/spindle_seal.html

And they're CHEEP, $12 for a kit to do both sides of the axle. I bought them online right from the company; http://www.bearingbuddy.com/orderss.htm