Would MacBook Air 2018 handle a constant run of a Windows VM for the Development My work involves having both Mac and Windows on a laptop.
While I need Mac for Xcode, I do some ASP.NET stuff tightly coupled with the Windows ecosystem.

I wonder if a MacBook Air CPU with 16 GB RAM would be enough to handle VMware Fusion with a couple of Visual Studio instances and debugging ASP.NET sites with the database on local MS SQL server?

There is an option of getting 13" MacBook Pro for a performance boost but I am frustrated with the touchbar as it's harder to use it with my fingers muscle memory. I'd like to avoid it as much as possible.
In general, MacBook Air's can handle run anything constantly, that they can handle running for a short while. You cannot "overwork" the computer so to say - it won't suddenly break or stop working for that reason.

And yes, MacBook Air's in general can handle running VMware Fusion (if the model you have is supported by VMware Fusion). You decide yourself how much RAM you dedicate to the virtual machine, when setting it up - so 16 GB will make it possible to run multiple virtual machines.

If you want to have a clearer idea of the amount of RAM needed, you'll need to setup the system you want and check how much RAM it uses.