I use an older Mac that I do not think will support the new version. How can I move Basecamp to a newer Mac that will support this new version?
I'm assuming you know that all you would need to do is to download the new version to a new Mac, but if you're asking about moving the data from the old Mac to the new one, then perform a backup.
I don't know how the Mac version works, but in Windows, one would open BaseCamp, click on "My Collection", go under "File", and then choose "Backup". Basecamp will ask for a location and name for the backup data. Once done, move the file to the new Mac, go under "File" and choose "Restore".