create bootable high Sierra for a mid 2011 Intel Mac mini.

In order to create a bootable installer, you will need a Mac that is compatible with High Sierra – one that is running some version of macOS now. If that old Mac mini is the only compatible Mac that you have, and you have no other bootable disks for it (such as clone backups), the only way to get to High Sierra directly would be through Internet Recovery – if that still works.

Use macOS Recovery on an Intel-based Mac - Apple Support

If you can get some version of macOS running on that Mac mini, you can use a modern Mac or PC to download the .DMG containing the Sierra installer, and put that on a USB drive as a data file (not as a bootable installer). Then you can transfer the Sierra installer to the Mac mini and install Sierra, which hopefully will bring the Mac mini up-to-date enough to be able to install High Sierra.

How to download and install macOS - Apple Support

Is the mac Mini bootable now? If so, what version of macOS is on it? Can you enter Recovery mode?

Downloading the High Sierra installer has proven to be difficult for many users. Best suggestion I can make is to read the article about it on MrMacintosh.

Unfortunately the M1 will not be usable for downloading & building a bootable installer for High Sierra.

Imnew2apple23 wrote:

The only Mac besides that Intel Mac mini I have is my M1 Mac mini that is running Golden Gate.

In that case, I don't think you're going to be making a bootable installer. Apple's instructions for creating one say that "In most cases, you must download from a Mac that is compatible with the macOS you're downloading."

Finding a used, original retail Mac OS X optical disk won't help you, either. Snow Leopard was the last version of Mac OS X that Apple distributed in that fashion, and that Mac mini cannot run anything earlier than Lion.

If memory serves correctly, Apple did distribute a few copies of Lion and Mountain Lion on USB flash drives, but the preloaded flash drives cost $80 and electronic downloads from the App Store cost $20. So the official USB flash drives probably would not have sold very well, and probably would be as scarce as hen's teeth now - even assuming that any of them still worked after all of this time.

Imnew2apple23 wrote:

The only Mac besides that Intel Mac mini I have is my M1 Mac mini that is running Golden Gate.

Follow @MartinR's instructions with the Mr Macintosh article. You will want to use Fix#3 in that article if you encounter a "server not available" error.

Boot the laptop using Command + Option + R to try boot to the online macOS 10.13 High Sierra installer. Unfortunately some Macs may instead only boot to the older online macOS installer (Lion, Mountain Lion) regardless of the keys used for booting. If this happens, then you will need to use Disk Utility to partition & format the whole physical internal drive using the instructions in the following article.....disregard the boot options in the following article....the important part is the use of Disk Utility to prepare the physical drive for installing of an older OS from macOS 10.7 Lion to macOS 10.10 Yosemite.

https://eshop.macsales.com/tech_center/formatting/Mac_Formatting_6-10.pdf