This information I am providing for those who are considering crossing the border from Zambia to Zimbabwe to experience fully all of Victoria Falls.  Since the chasm the falls created is so narrow, it is impossible to see all of it from the Zambia side.  You will probably see at most about 2/3 of the falls.
 
Our border crossing experience was without incident.  This was in July 2006, and from the perspective of a US citizen, though the person standing next to me was Finland, and he appeared to have no problems either.
 
After you cross the bridge that spans the gorge (the actual border being at the midpoint of the bridge), you walk perhaps half a mile to the actual border post.  You enter and stand in line after filling out the required paperwork- a single application form for a visa, one for each passport.  It was US$30 per person for a day pass at the time of this writing.
 
When you approach the customs agent, he will stamp the visa into your passport, annotate it, and give you an exit slip, which you need to hold onto and present when leaving the country.
 
From there, we walked to the entrance to Victoria Falls National Park, just up the road a few hundred meters further.  Afterwards, we took a taxi on to the Victoria Falls Hotel, who also returned us to the border crossing.  Leaving the country the same day was without incident, as was re-entry to Zambia.