Hi What2Dew,
If it turn out it’s not your door gasket thats causing the stream to leak out, then it sounds like you have a dishwasher along the lines as mine.
My dishwasher is a older model ASKO and in the top left side of the door on the outside is a small vent. When the dishwasher reaches the drying cycle a signal is sent to open flap inside the door to allow the steam to escape. I so far I only know of two other brands that have stream vents like mine and they are Miele and Frigidair.
When I first saw stream billowing out of the vent I thought the dishwasher was on fire. Below are a couple of ideas on how to seal the underside and seam of your kitchen counter.
http://www.diydoctor.org.uk/projects/pvaadhesive.htm
http://www.amesresearch.com/basement.htm?gclid=CL6WhNyX64YCFUf0JAodmUnSAw#b
Below is a picture of a Frididaire Dishwasher were you can see the vent in the top right hand corner. Is that along the lines of your Dishwasher?
Hope this helps you out,
Regards from Noelene
http://www.frigidaire.com/products/dishwashers/built-in_dishwashers/24_inch/prod_FDB126RB.asp