Ramadan, is no doubt, a blessed month; blessed by the Almighty with series of virtues. And that is what you mean by saying, Ramadan Mubarak; Blessed Ramadan. The Prophet (salla LLAAHU 'alayhi wa sallam) as well gave it that attribute in a Hadith that can be found in Sunan An-Nasaa'ee when he said,
'أتاكم رمضان شهر مبارك...'
'Ramadan has come to you, a _blessed month_...'
These show saying 'Ramadan Mubarak' is outrightly correct.
How about 'Ramadan Kareem'?
The phrase 'Ramadan Kareem' has been trailed by controversies, garnering criticisms from some set of individuals.
But what does it mean?
It has diverse meanings, the intended meaning determines its correctness or otherwise.
Kareem (كريم) is a word that comes with the وزن (measure) of فعيل, which, linguistically, may mean فاعل (an active particle) or مفعول (a passive particle).
That being said, kareem as an active particle (فاعل) can be translated as beneficient, benevolent. With this, Ramadan Kareem means, Beneficient Ramadan, and that makes it wrong. Ramadan isn't beneficient (it's just a month), Allah is.
Kareem as a passive particle (مفعول) means honoured. With that, Ramadan Kareem means Honoured Ramadan. Allah honours Ramadan, the believers as well honour Ramadan. Anything wrong with this? Absolutely, no.
Conclusively, Ramadan Mubarak is correct as well as Ramadan Kareem when the passive form of Kareem is intended (Honoured Ramadan).
Contemporary scholars with this opinion?
We have the likes of Shaykh Sulaiman bn SaleemuLlah Ar-Ruhayli and Shaykh Salih bn Hamad Al-'Usaimiyy (حفظهم الله).