Primarily, there is nothing wrong with wishing one another well at the start of or during Ramadan with the phrase ‘Ramadan Mubarak’ or ‘Ramadan Kareem,’ as such greetings are part of customs, and permissibility is the original rule governing such things.
Sheikh Muhammad Saleh Al-Munajjid, states the following: There is nothing wrong with felicitating one another well at the beginning of the month of Ramadan. The Prophet (peace and blessings be upon him) used to tell his Companions the good news of the onset of Ramadan, and he used to urge them to make the most of it. Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings be upon him) said, “There has come to you Ramadan, a blessed month. Allah has made it obligatory on you to fast (this month). During it, the gates of Paradise are opened and the gates of Hell are locked, and the devils are chained up. In it there is a night that is better than a thousand months, and whoever is deprived of its goodness is deprived indeed.” (Reported by an-Nasaa’i)
It has become clear now that felicitations during Ramadan are not considered an innovation or bid`ah because they are part of the custom of people and the act itself is based on the original rule of permissibility.