I downloaded the Oracle SQLDeveloper, but when I opened it, it said that it requires a minimum of Java 8 and gave me the website for the download. I went on and downloaded Java
If OpenJDK would work for you, and you are already using/willing to use Homebrew, AdoptOpenJDK works for me:
brew tap AdoptOpenJDK/openjdk
brew install adoptopenjdk8
I have not tested this thoroughly, but at the very least I am able to bring up SQL Developer 19.1.0.094, connect to a database, and run queries on it.
It did not work if I tried to use their instructions to install the "latest" version: brew cask install adoptopenjdk, as this seems to install OpenJDK 12.x as of May 2019.
There are other methods to install OpenJDK, but this was an easy one.