Beta system software should not be run on production machines. Often strange problems occur. It is quite likely caused by bugs in the Mojave beta.
...or incompatibilities introduced by new features or debugging stuff or similar.
Using a beta OS to run standard software should only be done when adviced doing so.
But it’s good to know...so may be you should have added more detail:
- Indesign version (exact version numbers)
- Machine configuration.
The software may run well in other configurations.
My Indesign Crash! Not open... I have installed Mojave 10.14
Process: Adobe InDesign CC 2018 
Path: /Applications/Adobe InDesign CC 2018/Adobe InDesign CC 2018.app/Contents/MacOS/Adobe InDesign CC 2018
Version: 184.108.40.206 (13100)
Code Type: X86-64 (Native)
Parent Process: ??? 
Responsible: Adobe InDesign CC 2018 
User ID: 501
Date/Time: 2018-07-19 19:10:01.613 -0300
OS Version: Mac OS X 10.14 (18A336e)
Report Version: 12
Bridge OS Version: 3.0 (14Y664)
Anonymous UUID: B5CC58B4-2F7E-21B3-96C2-D5ECBB034D19
[Crash report removed by moderator]
This is undoubtedly a Mojave bug. Send your reports to Apple. If you're a beta tester you have a link for doing that.
Also, do not post full crash reports here. Post them on Pastebin.com and provide a link.
Here's Adobe's response:
Despite the fact that Adobe seems to be not too keen on testing Betas and supporting Public Betas, maybe it helps the devs to have some feedback on Application behaviour.
Running the Beta on a MacBookPro to see how new features work, I ran into a crash-behaviour with Indesign.
MBP, 13" Touch Bar, late 2016, i7 3,3, 16 GB, Iris 550 Graphic.
Latest Public Beta of Mojave, build 18A336e.
Adobe CC Indesign 2018 (latest version)
Indesign crashes upon launch, invariably, no fix found so far. Other CC-Apps seem to work. Could try to disable SIP or re-install, but wanted to inform you first.
I think always helps to get devs a heads-up (here and via the feedback site), same goes for Apple (and other apps having issues).