Praise be to Allah.
Part of the etiquette of reciting Qur’aan is for the reciter to always say Bismillah ir-Rahmaan ir-Raheem (In the name of Allah, the Most Gracious, the Most Merciful) at the beginning of every soorah, except Soorat at-Tawbah.
Imam an-Nawawi (may Allah have mercy on him) said: He should always recite Bismillah ir-Rahmaan ir-Raheem at the beginning of every soorah except Soorat Baraa’ah (at-Tawbah), because the majority of scholars said that it is a verse wherever it is written in the Mushaf, and it was written at the beginning of all the soorahs except Soorat at-Tawbah. If he recites it at the beginning of every soorah (other than at-Tawbah), then there is certainty that he had recited the entire Qur’aan or soorah, and if he fails to recite the Basmalah, he has omitted part of the Qur’aan according to the majority.
End quote from at-Tibyaan fi Adaab Hamalat al-Qur’aan, 100.
But if he recites from partway through a soorah, he does not have to recite the Basmalah and it is sufficient for him to seek refuge with Allah (by saying A‘oodhu Billaahi min ash-Shaytaan ir-rijeem (I seek refuge with Allah from the accursed Shaytaan), but if he does say the Basmalah there is nothing wrong with that. For more information please see the answer to question no. 21722
The reason why the Basmalah does not appear at the beginning of Soorat at-Tawbah is that the Sahaabah (may Allah be pleased with them) did not write it at the beginning of this soorah in the Mushaf, following the example of the caliph ‘Uthmaan (may Allah be pleased with him). At-Tirmidhi narrated in as-Sunan that Ibn ‘Abbaas (may Allah be pleased with him) said: “I said to ‘Uthmaan ibn ‘Affaan: ‘What made you put al-Anfaal which is one of the Mathaani, next to Baraa’ah [al-Tawbah], which is one of the Mi’een? Why did you not put the line Bismillaah ir-Rahmaan ir-Raheem in between them when you put it at the beginning of the rest of al-Sab’ al-Tiwaal [the long seven soorahs]?” ‘Uthmaan said: “The Messenger of Allaah (peace and blessings of Allaah be upon him) used to receive revelation of soorahs with many aayahs. When they were revealed, he would call his scribes and tell them, ‘Put these aayahs in the soorah where such-and-such is mentioned.’ Al-Anfaal was one of the first soorahs to be revealed in Madeenah, and Baraa’ah (al-Tawbah) was one of the last parts of the Qur’aan to be revealed. Its stories were similar to the stories mentioned in al-Anfaal, so it was thought that it was part of it. The Messenger of Allaah (peace and blessings of Allaah be upon him) was taken [i.e., died] without explaining whether it was indeed part of it, so they were put next to one another, and the line Bismillaahi ir’Rahmaan ir’Raheem was not written between them, and it [al-Tawbah] was put among the Sab’ al-Tiwaal [seven long soorahs].”
End quote from Fataawa al-Lajnah ad-Daa’imah, 4/225
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: … The correct opinion is that there is no Basmalah between it (Soorat at-Tawbah) and al-Anfaal, because the Basmalah is a verse of the Book of Allah, may He be glorified and exalted, and if the Messenger (blessings and peace of Allah be upon him) did not say to put the Basmalah between the two soorahs, they did not put it between them. The Prophet (blessings and peace of Allah be upon him) is the one who decided and said: Put the Basmalah, and he did not tell them to put the Basmalah between Soorat al-Anfaal and Soorat at-Tawbah, so they did not write it. But it remains to be said: if he did not give instructions to this effect, why is there a separation between it and Soorat al-Anfaal? Why did they not make it one soorah? We say: Yes, they did not make it one soorah because they were uncertain as to whether it was the same soorah as al-Anfaal or they were two separate soorahs, so they said: Let us make a separation between the two soorahs and not write the Basmalah. This is the correct opinion concerning the absence of the Basmalah between Soorat al-Anfaal and Soorat at-Tawbah.
End quote from Liqa’ al-Baab al-Maftooh, no. 18
And Allah knows best.