carestad

If your site is depending on posts to have their thumbnail images set, this is the plugin for you. Also makes it possible 
to require tags in every new post.
When trying to save a post without at thumbnail or tags and when activated, the user will get a warning and the post will 
be saved as a draft instead.

Post Requirements

This plugin allows you to require posts to have thumbnail set and/or tags.

Jatinder Pal Singh

This is a simple plugin which is designed to help the posters who some times forget to select the post category and post is published under default category. Once activated, this plugin will force the users to select post category while adding new post to wordpress site. Whenever User tries to publish a post without selecting category, it will give an alert to the user and also highlight the category selection area so that user can see from where to select the post category. 
Moreover, If poster hasn’t selected the category while clicking on publish button, it will save the post as draft under default, so no need to worry about loosing the post content in any case due to this plugin.

Force Post Category Selection

Just found this little gem of a plugin. Its so simple to use, literally just install then forget about it. I have two categories, and needed to make sure at least one of them was selected. This did the job perfectly. Thanks!

Install, then forget. A+

This is simple plugin which forces the users to select atleast one category from the list while publishing a new post.

Post Requirements vs Force Post Category Selection