Ramadan Mubarak is the most common greeting used by Muslims, and translates to “have a blessed Ramadan”.
Ramadan Kareem is another phrase often used, however there is some debate as to whether it is appropriate.
“Ramadan Kareem” means (generous ramadan for you), it’s new to our era, it wasn’t used in old Islam times.
Some people see it’s wrong, as Ramadan is a month, because you obviously cannot describe a month with generosity, and you can’t say it’s generous to its people is it isn’t the one who gives, but Allah swt who gives and he is who’s very generous and merciful.
Some others thinks the complete opposite of the above, they thinks it’s the month of generosity, as Allah opens the gates of the heavens so that our Duaa’s are granted, and he doubles our ‘Good deeds’ Ten times!
And “Ramadan Mubarak” means (Blessed Ramadan), Prophet Muhammad PBUH used to say it to his people.
Regarding this topic, Saudi Arabian scholar Shaykh ibn-al'Uthaymeen said:
The ruling of that is that the statement 'Ramadan kareem' is not correct. Rather it should be said 'Ramadan Mubarak', or whatever is similar to it, because it is not Ramadan itself that gives so that it can be kareem (generous), in fact it is Allah who placed the grace in it, and made it a special month, and a time to perform one of the pillars of Islam.
It is as if the one saying this thinks that the sacredness of the time allows him to commit transgressions. This is turning around what the people of knowledge have said that bad deeds are greater at special times and places, the opposite of what the one saying this imagines. They say that it is obligatory to fear Allah - the honourable and majestic - at all times, and in all places, especially in special times and special places.
Allah - the honourable and majestic - says,
O you who believe! Observing As-Saum (the fasting) is prescribed for you as it was prescribed for those before you, that you may become Al-Muttaqun (the pious - see V.2:2). [2:183]
Allah shows that the wisdom of fasting is taqwa (piety) of Allah - the honourable and majestic - by fulfilling His commandments and avoiding His prohibitions. It has been ascertained from the Prophet (pbuh) that he said, "Whosoever does not leave speaking falsehood, or acting with it or ignorance, then Allah has no need that he leaves his food and drink." Fasting is cultivation of the self, guarding it from the prohibitions of Allah, and not as this ignoramus says that due to the sacredness and blessing of this month it allows commiting transgressions.
From a collection of the Shaykh Ibn al-'Uthaymeen's fatwas, As`ilah wa ajwibah 'an alfaaz wa mafaaheem fee meezaan al-sharee'ah ("Questions and Answers on Expressions and Concepts in the Scale of the Shariah") printed by Maktab al-Da'wah of Britain.
Also, another scholar Shaykh Fawzan said:
This has no basis. Their statement Ramadhan Kareem has no basis. As for “Ramadhan Shareef”, “Ramadhan Mubaarik” and “Ramadhan Atheem”, these descriptions appear in various narrations. Atheem, Mubaarik, Shareef, there is no problem (with these terms). As for Kareem, then I do not know of any origin for this.
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What is more appropriate as a Ramadan greeting? Ramadan Mubarak or Ramadan Kareem? - Quora