The instance I am on has upgraded to support mastodon public search. If (and only if!) you don't have a reason to be uncomfortable with being easily searchable, I recommend you turn it on as you never know when and how it will help someone looking for the answer to a very specific question, or trying to track down a fond memory of you.
These are the instructions that came with the update: "To enable search for your posts, go to settings -> Public Profile then select the Privacy and Reach tab and scroll down and check the box next to "Include public posts in search results", then click the save button." Settings is labeled "Preferences" for me.
(Your instance may not have updated yet or may never support it)