There’s been a discussion on GitHub about this problem https://github.com/schellingb/dosbox-pure-unleashed/issues/2
It turns out that on recent versions of macOS the operating system will put anything downloaded into a quarantine so downloaded programs can’t be run. To remove said quarantine-flag you will need to run a command in the terminal:
xattr -dr com.apple.quarantine /Applications/DOSBoxPure.app
After that it should run. I now wrote this information into the documentation into the Download section.