Drop shadow greyed out

Please do a test on vector shape to provide us more information. And as already mentioned make sure you are using the selection tool to select type as an object, not text tool which selets as text character(s). 

OS & System match

What version of OS & Illustrator are you using? Make sure they are compatible with each other.

Illustrator system requirements | earlier releases

Have you restarted or reset your prefs? Substitute 22 for your version.

PREFS  RESET (thorough method)

Quit Illustrator

Finder >> Shift Command G

~/Library/Preferences/Adobe Illustrator 22 Settings/en_US*/Adobe Illustrator Prefs

rename Adobe Illustrator 22 to Adobe Illustrator 22 old

Here are a few suggestions:

1. Make sure you're using the latest version of Illustrator.
2. Reset preferences: https://helpx.adobe.com/illustrator/kb/troubleshoot-preference-issues.html
3. Uninstall/reinstall Illustrator.
4. Use the Creative Cloud Cleaner Tool: https://helpx.adobe.com/creative-cloud/kb/cc-cleaner-tool-installation-problems.html

How do you select the text?

Don' use the text tool.

You need to select the text object by clicking on it once (!) using the selection tool.