The eminent Muslim scholar and renowned Da`iyah, Sheikh `Abdel Khaliq Hasan Ash-Shareef, states: “The whole issue is controversial among Muslim jurists. Muslim scholars differed whether basmalah is a verse of each surah [exceptsurat At-Tawbah] or not. Some scholars consider it as a verse of the Qur’an while others do not consider it as a verse. Those who consider it as a verse differed whether it should be recited loudly in the loud Prayer or it can be recited silently. As the issue is controversial among Muslim jurists, a Muslim is allowed to follow the opinion that best suits him.

Thus, if he recited Al-Fatihah without basmalah, a group of Muslim scholars are of the opinion that his Prayer is valid. However, it is preferable if he recited the basmalah silently or loudly before Al-Fatihah. Many people recite the basmalah silently before reciting Al-Fatihah and there is nothing wrong in that.”

Elaborating more on this issue, we’d like to furnish you with the following fatwa issued by the lateSheikh `Atiyyah Saqr, former head of Al-Azhar Fatwa Committee who stated:
“Scholars have unanimously agreed that the Basmallah, or the phrase “in the Name of Allah, Most Gracious, Most Merciful”, is part of a verse in the Qur’an, namely the verse in surat An-Naml that reads: “It is from Solomon, and is (as follows): In the name of Allah, Most Gracious, Most Merciful.” (An-Naml: 30)
however, Muslim scholars differ as to whether or not it constitutes a verse of every surah (chapter) in the Qur’an. There are three opinions regarding this point:
The first view:
Basmallah stands as one of a verse of surat Al-Fatihah and of every surah of the Qur’an. Therefore, it has to be recited with Al-Fatihah during those prayers of loud or inaudible recitation. The strongest support of this opinion comes from the hadith of Nu`aym Al-Mujammir who said, “I prayed behind Abu Hurayrah and he recited, ‘In the Name of Allah…’ and then he recited Al-Fatihah.” At the end of the hadith, he is quoted as saying, “(I swear) By the One in whose Hand is my soul, I have prayed the same way the Prophet (peace and blessings be upon him) used to pray.”
The second view:
It is a verse by itself and was revealed to demarcate different surahs. Hence, it is allowed to recite it with Al-Fatihah (in fact it is preferred), but it is not to be said aloud. The proof of this view is the hadith reported by Anas (may Allah be pleased with him): “I prayed behind the Messenger of Allah, Abu Bakr, `Umar and `Uthman, and they did not recite it aloud.”
this hadith is related by An-Nasa’i, Ibn Hibban and At-Tahawi with a sahih chain of narrators.
The third view:
It is not a verse of Al-Fatihah or of any other surah in the Qur’an. It is disliked to recite it aloud or quietly during obligatory prayers, not the supererogatory
ones. This opinion, however, does not have a strong proof.
We can conclude that sometimes the Prophet would recite Basmallah aloud, and some other times he would say it quietly. So long as the matter is controversial, we should not stick to one view and discard other views. I would say it is better to do all of them, and not doing it at all does not render one’s prayer null and void.”

Given the above, it becomes clear that if one forgot to say basmalah before reciting Al-Fatihah, there is nothing wrong with his Prayer and he does not need to recite Al-Fatihah again. However, it is preferable to be keen to recite basmalah before Al-Fatihah either silently or loudly. May Allah guide you and us all to the right path, amen!