In 2022, you can use just brew

brew install openjdk

and maybe you need to update PATH env:

export PATH="/usr/local/opt/openjdk/bin:$PATH"

…and for the future give a try to sdkman, is better than brew

curl -s "https://get.sdkman.io" | bash

then open a new shell and try list to see what you could install ;-)

sdk list java 

At time of writing you could use:

sdk install java 17.0.1-tem

To entirely use a (bash) shell to install e.g. Java 8u112 JRE do the following:

Download:

cd ~/Downloads
curl -v -j -k -L -H "Cookie: oraclelicense=accept-securebackup-cookie" http://download.oracle.com/otn-pub/java/jdk/8u112-b15/jre-8u112-macosx-x64.dmg > jre-8u112-macosx-x64.dmg

Mount:

hdiutil attach jre-8u112-macosx-x64.dmg

Install:

sudo installer -pkg /Volumes/Java\ 8\ Update\ 112/Java\ 8\ Update\ 112.app/Contents/Resources/JavaAppletPlugin.pkg -target /

Clean-up:

diskutil umount /Volumes/Java\ 8\ Update\ 112 
rm jre-8u112-macosx-x64.dmg

To install the latest Oracle JDK (at the time of writing this answer it's Java 8u121 JDK) the various paths (besides the version number) are slightly different compared to the older JRE version. The URL contains an additional directory (i.e. /e9e7ea248e2c4826b92b3f075a80e441) and the dmg file contains a .pkg file instead of an installer app:

cd ~/Downloads
curl -v -j -k -L -H "Cookie: oraclelicense=accept-securebackup-cookie" http://download.oracle.com/otn-pub/java/jdk/8u121-b13/e9e7ea248e2c4826b92b3f075a80e441/jdk-8u121-macosx-x64.dmg > jdk-8u121-macosx-x64.dmg
hdiutil attach jdk-8u121-macosx-x64.dmg
sudo installer -pkg /Volumes/JDK\ 8\ Update\ 121/JDK\ 8\ Update\ 121.pkg -target /
diskutil umount /Volumes/JDK\ 8\ Update\ 121
rm jdk-8u121-macosx-x64.dmg

Oracle has a poor record for making it easy to install and configure Java, but using Homebrew, the latest OpenJDK (Java 14) can be installed with:

brew install --cask adoptopenjdk8

For the many use cases depending on an older version (commonly Java 8), the AdoptOpenJDK project makes it possible with an extra step.

brew tap adoptopenjdk/openjdk
brew install --cask adoptopenjdk8

Existing users of Homebrew may encounter Error: Cask adoptopenjdk8 exists in multiple taps due to prior workarounds with different instructions. This can be solved by fully specifying the location with brew install --cask adoptopenjdk/openjdk/adoptopenjdk8.

I tested this answer using a clean install of High Sierra 10.13.6. All updates from Apple were installed.

The downloads of the JDKs can be found at Java Downloads. I could not correctly view this site using Safari under High Sierra. Instead, I installed and used Firefox ESR (Extend Support Release). This link for Firefox download can be successfull viewed from Safari. The file I downloaded and installed was Firefox 115.16.1esr.dmg.

I tried JDK 23 by downloading and installing jdk-23_macos-x64_bin.dmg. This was not recognized by Java. Next, I tried JDK 21 by downloading and installing jdk-21_macos-x64_bin.dmg. This was recognized by Java.

Instructions for uninstalling a JDK can be found here. Personally, I just used the Finder to move the /Library/Java/JavaVirtualMachines/jdk-23.jdk folder to the Trash.

I tested by using the simple code given below. The code was stored in the file named Simple.java.

class Simple{  
    public static void main(String args[]){  
     System.out.println("Hello Java");  
    }  
}

Example of testing is given below.

dmas-Mac:Desktop dma$ javac Simple.java 
dmas-Mac:Desktop dma$ java Simple
Hello Java

BTW, I tested JDK 23 using macOS Monterey and did not encounter the problems that occurred with High Sierra.