Ramadan is not kareem, but your Rabb is indeed Kareem, Subhanahu wa ta’ala.
The ruling concerning this is, this phrase ‘Ramadan Kareem’ is not correct, and the only phrase that should be said is ‘Ramadan Mubarak’ or what resembles that. Because Ramadan is not the one that gives such that it can be called generous (Kareem), rather it is only Allaah, the Exalted, that put virtue in it and made it a virtuous month and made the time period for performing (the fasting) a pillar from Islam.
Al-Kareem (The Generous One) is one of the Beautiful Names of Allaah, the Exalted and it is among His attributes. It is incumbent upon the person to have Taqwa of Allaah the Mighty and Majestic during every time and in every place especially during virtuous times and in noble places.
And Allaah the Mighty and Majestic says:
“O you who have believed, decreed upon you is fasting as it was decreed upon those before you that you may become righteous -” {Quran, 2:183]