I'm not an expert on this (see my previous posts for evidence
) but I think it's pretty much the standard to open the channel (I'm assuming Serial/RS485 or similar) and keep it open.
Opening the COM port and establishing a connection is orders of magnitude slower than sending actual traffic once the port is open. Also, if you want to read information via MODBUS then you'll be polling the COM port many times per second.
With regard to saving computer time, the processing power needed to maintain a serial connection is negligible to any modern computer and by modern, in this case I mean any PC less than 20 years old.
If you're having regular communication port failures then it could be that you've a lot of electrical noise in your system and it could be worth trying to resolve that. VFDs can be a particular source of electrical noise. In my experience the popular low cost Huanyang brand of VFDs were incredibly noisy, I gave up trying to resolve that with shielding and grounding and just replaced them with other brands.
Hope this helps