How do I run the ViennaRNA package according to the MacOS X distribution.
It goes into the Downloads folder and then Nothing happens?
Let's take a step back. The ViennaRNA package is used for RNA secondary structure prediction (generally speaking) and not for primary structure/sequence analysis operations such as edit distance or hamming distance calculations. What exactly do you want to do?
To directly address your problem (regardless of whether it enables you solve your original task): I assume you have downloaded the installer from https://www.tbi.univie.ac.at/RNA/#download? That one says "Failed - No file", so I think the link is broken and you might want to look for a contact on that website to get it fixed. Given your problem description "I want to use RNAmute RNAfold, RNAsubopt to calculate the edit distance and hamming distance of my sequence.", I think this will not help you, though.
edit: fixed grammar mistake
My advice would be download Conda and install it from the relevant channel. Pretty sure I did this in the past, but cschu181 raises valid points that should be addressed first.