This is a longe debated question, please have a look at this webpage for some pros and cons for each side.
We stayed on Zambia side and just went to Zim side for an afternoon. I think the Zambia side is much cheaper and there a lot of cool adventure (like bungee jumping etc) activities to do on the Zambia side. We did the cheetah interaction which was very fun. The Falls in June 2013 were much better on the Zambia side so I would stay there. You can take a taxi to get out to the Falls. Zimbabwe is ruled by a dictator who skims off all the visa money etc for himself so I wouldn't want to contribute to his coffers. I can recommend the Green Tree Lodge which was about $80/night. Jolly BackPacker or something like that was the option for cheaper accommodation.
Just to clarify in order for you to make an informed desicion on the correct information. Zambia on the whole is actually MORE expensive than Zimbabwe, plus if you are staying in any of the Livingstone lodges like Green Tree Lodge or Jollyboys, transfers to the activities are not included which means you need to add on transfer or taxi costs. In Zimbabwe because the town is situated so much closer to the Falls collection and drop off from your hotel is included in the activity price.
Adventure activities like the bungi jumping can be done from either country, and most activities are available on either side except for the cheetah interaction, microlighting and the Livingstone island tour/ Devils swim which are only available in Zambia.
The views of the Falls are lovely from Zambia and especially from Livingstone island, but it is important to note that 70% of the Falls are on the Zimbabwean side and anytime from late September through to early December the Falls in Zambia have just about completely dried up.
Visa fees have to be paid for in both countries and I am not sure that it is a fair statement to say that in Zimbabwe these go straight into the Presidents pocket.
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