公共の場で喫煙は控えてください。

English Translation

Please refrain from smoking in public places.

公共の場で喫煙はお控え下さい

sounds more natural than 控えてください although 控えてください isn’t bad.

Inserting の makes the whole sentence meaning “you are okay to smoke somewhere else, but not here”. So, the speaker narrows down the prohibited area for smoking by inserting の.

お控え下さい is a common phrase for a notification or poster in a public space. It may look polite because of ください, but actually 控えてください isn’t polite enough in this situation.