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

Turns out java has been moved into brew core recently, so the correct command as of August 2022 is:

brew install java

Then check your installation by running

java -version

If the result does not looks like this:

openjdk 18.0.2 2022-07-19
OpenJDK Runtime Environment Homebrew (build 18.0.2+0)
OpenJDK 64-Bit Server VM Homebrew (build 18.0.2+0, mixed mode, sharing)

but like this:

The operation couldn’t be completed. Unable to locate a Java Runtime.
Please visit http://www.java.com for information on installing Java.

Then you also need to create a symlink for the system Java wrappers to find this JDK:

sudo ln -sfn /opt/homebrew/opt/openjdk/libexec/openjdk.jdk \
     /Library/Java/JavaVirtualMachines/openjdk.jdk

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.