All perfect praise be to Allah, The Lord of the Worlds, and may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
There is no harm in congratulating each other by saying "Ramadan Kareem" to show happiness and joy over the start of this blessed month. This is provided that there is no Sharia-approved evidence prohibiting that. Moreover, associating this phrase with Ramadan reflects the feature of "Karam"(Generosity) inherent in it since, during this month, Allah, The Almighty, rewards His slaves generously and bestows his favors and blessings upon them abundantly. Allah Said (what means): "(The queen) said: "Ye chiefs! here is delivered to me - a letter worthy of respect." (An-Namel/29). He, The Exalted, Said as well (What means): "We did, before them, try the people of Pharaoh: there came to them an apostle most honourable," (Ad-Dukhan/17).
In conclusion, there is no harm in congratulating each other by the above phrase, and we believe that there is no evidence in Sharia prohibiting that. And Allah knows best.