Does Mathematica 11 work with MacOS Catalina 10.15
As already stated elsewhere, versions 11.3 and earlier simply cannot run on Catalina because Apple removed support for 32-bit applications. And, unfortunately, creating a 64-bit 11.3.1 impossible, because the 11.3 FrontEnd is a "Carbon" application, rather than a "Cocoa" application. We've been working on rewriting the FrontEnd in Cocoa for several years, but it was not a simple or easy project (especially when you have 30-year old Macintosh code, such as QuickDraw, hiding in various places). Whole sections of the FrontEnd had to be thrown out and rewritten from scratch.
One solution that does not involve upgrading Mathematica or downgrading macOS is to use a virtual machine to run Mathematica. You can use a program such as Parallels, VMWare Fusion, or (the free) Virtual Box to create a virtual machine running macOS Mojave. Mathematica runs well in a virtual machine--I regularly do so for testing purposes. And as long as you're not doing anything disk-intensive (and definitely most Mathematica operations are not disk-intensive), the performance cost of using a VM is negligible.
No, unfortunately there are no easy workarounds - MacOS Catalina simply doesn't support 32-bit apps, and the Mathematica frontend for 11.3 and earlier is a 32-bit app.
To solve this, you must either upgrade your license or downgrade your version of MacOS. You may also have some luck contacting Wolfram Support and seeing if they can help you.
I received a response from Wolfram Cutomer Support today regarding this question. The answer was to update to Version 12 !!! I think this is completely shameless. I have Version 11.3, which is Wolfarm's second most recent version of its software, which it apparently refuses to update even though Apple has long announced that 32-bit programs will not work. This is likely in an attempt to force customers to update to their latest version. This is the most shameless thing I've seen in the software industry. I have a relatively large number of other software in my computer (both newer and older versions) and NO other software has any problem with the Mac OS update.