You will need to remove the lower kick panel directly below the door to gain access to the inside cabinet to see where the leak may be coming from. The panel comes off by removing the screws under the bottom front edge of the panel. With the screws removed, the panel will drop down, then come off.
NOTE: Propping the two front feet of the washer up with something (a 2x4 works well for this) will give you better access to the screws.
The most common source of leakage on this washer is the rubber bellow around the door. It can develop cracks or tears over time and require replacement. A tell-tale sign is the front counterweight (looks like a concrete block on the front of the wash tub) looking wet. The counterweight will remain dry if the tub is sealed properly. From the inside of the washer, pull the rubber folds back on the door bellow and inspect the perimeter for any rips or tears. If the bellow is damaged, it must be replaced. The following link epxlains how:http://www.fixya.com/support/r385210-replacing_door_bellow_kenmore_elite
If the bellow is intact, check the rear bellow seal where it connects to the front of the wash tub. If the clamp comes loose the bellow can shift and begin to leak. Tighten if necessary.
Check the drain pump and hoses to ensure they are not leaking. The drain pump is directly in the front under the wash tub.
Check the area directly below the wash tub. If the leak is coming from the wash tub iteself, you may have a bad tub seal. A replacement gasket doesn't cost much at all, but the labor involved is extensive.
Another simple problem is the dispenser tube(s) that come through the door bellow. Sometimes, these small tubes will work their way behind the rubber in the door and leak when the washer attempts to fill or rinse. This is a simple matter of adjusting the tubes to ensure they are seated through the door gasket properly.
If you have any questions, or require additional assistance, please let me know. I hope you find this information helpful.
NOTE: The repairman was correct about leaving the door open to prevent mold and mildew. Due to the water tight seal of these doors it is important to leave the door open when not in use to prevent any residual water from becoming stagnant and allows the wash tub to air out. Also, you should always check and clean the area around the door. Small articles of clothing can get stuck in this area and begin to mildew as well. I have repaired this washer numerous times and personally own one. I have never had a problem with it in the 9 years I've had it, following this regiment.