Sheikh Sâlih al-Fawzân
There is a general supplication that should be recited upon entering upon any month of the year. The Prophet (peace be upon him), when he saw the first crescent Moon that heralds the arrival of a new month, would say: “O Allah! Make it emerge over us with prosperity and faith, with safety and Islam. My Lord and the Lord of (this month) is Allah.” [Sunan al-Tirmidhî (3451)]
Allâhumma! Ahlilhu `alaynâ bil-yumni wal-îmân, was-salâmati wal-Islâm, Rabbî wa Rabbuka-llâh.
In some narrations it reads: “O Allah! Make it emerge over us with security and faith, with safety and Islam. My Lord and the Lord of (this month) is Allah.” [Mustadrak al-Hâkim (4/285 #7837)]
Allâhumma! Ahallahu `alaynâ bil-amni wal-îmân, was-salâmati wal-Islâm, Rabbî wa Rabbuka-llâh.
This is the supplication that is found in the Sunnah for sighting the new moon, whether it is on the occasion of Ramadan or any other month.
As for a supplication that is prescribed specifically for Ramadan, I do not know of any. However, if a Muslim asks Allah to assist him in fasting the month of Ramadan and to accept it from him, then there is nothing wrong with this, as long as a specific wording is not adhered to for that purpose.
Source: Islam Today