Praise be to Allah.
Reciting the Basmalah (i.e., the phrase “Bismillaah ir-Rahmaan ir-Raheem (in the name of Allah, the most Gracious, the Most Merciful)” is mustahabb (encouraged), not obligatory. It is mustahabb if you are going to recite a soorah from the beginning, so the worshipper says it before al-Faatihah. With regard to the soorah after al-Faatihah, if he is reciting it from the beginning of the soorah, then he should say the Basmalah, apart from Soorat al-Tawbah, because the Basmalah is not recited at the beginning of it. If he is reciting from the middle of a soorah, then it is not mustahabb to recite the Basmalah.
The Standing Committee said:
The sound Sunnah indicates that the Prophet (peace and blessings of Allah be upon him) recited the Basmalah in prayer before al-Faatihah and before other soorahs, apart from Soorat al-Tawbah, but he did not recite it out loud in the prayers in which he recited Qur’aan out loud.
Fataawa al-Lajnah al-Daa’imah, 6/378
They also said:
It is prescribed to recite the Basmalah in every rak’ah before al-Faatihah and before every soorah apart from Soorat Baraa’ah (i.e., Soorat al-Tawbah).
And they said:
If (the worshipper) is going to recite a soorah after al-Faatihah, he should recite the Basmalah silently beforehand. But if he is going to recite a brief passage from the middle or end of a soorah, then it is not prescribed for him to recite the Basmalah.
Fataawa al-Lajnah al-Daa’imah, 6/380
And Allah knows best.