As per Chapter 55, Rudra Samhita (Yuddha Khanda) of Shiva Purana and Chapter 63, Book 10 of Srimad Bhagvatam, Sri Krishna only cut hands of Banasura. But Chapter 12, Part 1 of Agni Purana doesn't mention cutting of arms . Chapter 56, Rudra Samhita (Yuddha Khanda) of Shiva Purana also says Bana acquired position of a Gana in Kailasa.
As per Shiva Purana, Banasura did tapasya to Shiva to get rid of his extra arms which was aquired by Tapasya previously. Lord Shiva says his arms will be cut by a person who is equal in valor to Him (Shiva).
Chapter 55 of Yuddha Khanda says:
ततः सुदर्शनेनाशु कृष्णो बाणभुजान्बहून्।
Then lord Krsna, the destroyer of heroic armies, at the bidding of
Siva, suddenly cut off many arms of Banasura, with the Sudarsana
अवशिष्टा भुजास्तस्य चत्वारोऽतीव सुन्दराः।
शङ्करस्य प्रसादतः॥ ३१॥
By the grace of Siva, only four beautiful arms were left out and soon
his misery was over.
Chapter 10.63 of Bhagavatam also says the same with slight variation.
दर्पोपशमनायास्य प्रवृक्णा बाहवो मया
सूदितं च बलं भूरि यच्च
भारायितं भुवः ॥ १०.६३.४८ ॥
It was to subdue Bāṇāsura’s false pride that I severed his arms. And I
slew his mighty army because it had become a burden upon the earth.
Chapter 56 Yuddha Khanda (Rudra Samhita) of Shiva Purana says Shiva bestowed him position of Gana.
इति श्रुत्वा स्वभक्तस्य बाणस्य भगवान्भवः।
लभिष्यसीत्युक्त्वा तत्रैवान्तरधीयत॥ ३२॥
Thus listening to the prayer of Banasura, lord Siva said, “You will be
benefited enormously.” Thus speaking he disappeared from the scene.
ततः शम्भोः प्रसादेन महाकालत्वमागतः।
Then with the grace of lord Siva, Bana attained the immortality of the
eternal times and became one of the gana of Siva, with delight.
But Chapter 11 Rudra Samhita (Kumara Khanda) of Shiva Purana and Gaguli's edition of Chapter 46, Salya Parva of Mahabharata say Kartikeya (Kumara) killed Banasura. But BORI edition of Mahabharata omits this line.
After Killing of Tarakasura, Kumara swamy was on rage and he started killing all daityas on earth with Shakti weapon to reduce burden on earth. Bana, son of Mahabali, in fear hid into Krauncha mountain and Kumara shattered the mountain with his javelin.
Shiva Purana says Krauncha mountain prayed to Kartikeya to relieve him from Banasura while Mahabharata doesn't mention any prayer.
Chapter 11, Rudra Samhita (Kumara Khanda) of Shiva MahaPurana says:
चिक्षेप तां समुद्दिश्य स बाणं शङ्करात्मजः।
जज्वलुश्च दिशो नभः।। ९ ।।
Then the son of Siva, shot the Sakti over Banasura creating enormous
sound, as a result of which the sky with all the ten directions
started burning, creating the thundering sound.
सबलं भस्मसात्कृत्वाऽसुरं तं क्षणमात्रतः।
गुहोपकण्ठं शक्तिः सा जगाम परमा मुने!।॥ १०॥
O Sage, the Sakti shot by Kumāra,
reducing Banasura and his army to ashes in a moment, it then returned to Kumara.
ततः कुमार प्रोवाच क्रौञ्चं गिरिवरं प्रभुः।
निर्भयः स्वगृहं गच्छ
नष्टः स सबलोऽसुरः॥ ११॥
Kumara then said to the excellent Kraufica, "Now you should go back to
your home fearlessly, because the demon has been destroyed with his
Ganguli Edition describes the incident as follows:
Bali's son Vana of great might, getting upon the Kraunca mountain,
battled with the celestial host. Possessed of great intelligence, the
great generalissimo Skanda rushed against that foe of the gods. From
fear of Kartikeya, he took shelter within the Kraunca mountain.
Inflamed with rage, the adorable Kartikeya then pierced that mountain
with that dart given him by Agni. The mountain was called Kraunca
(crane) because of the sound it always produced resembled the cry of a
crane. That mountain was variegated with shala trees. The apes and
elephants on it were affrighted. The birds that had their abode on it
rose up and wheeled around in the welkin. The snakes began to dart
down its sides. It resounded also with the cries of leopards and bears
in large numbers that ran hither and thither in fear. Other forests on
it rang with the cries of hundreds upon hundreds of animals. Sharabhas
and lions suddenly ran out. In consequence of all this that mountain,
though it was reduced to a very pitiable plight, still assumed a very
beautiful aspect. The vidyadharas dwelling on its summits soared into
the air. The kinnaras also became very anxious, distracted by the fear
caused by the fall of Skanda's dart. The daityas then, by hundreds and
thousands, came out of that blazing mountain, all clad in beautiful
ornaments and garlands.
The followers of Kumara, prevailing over them in battle, slew them
all. The adorable Skanda, inflamed with rage, quickly slew the son of
daitya chief (Bali) along with his younger brother, even as Indra had
slain Vritra (in days before). The slayer of hostile heroes, Agni's
son, pierced with his dart the Kraunca mountain, dividing his own self
sometimes into many and sometimes uniting all his portions into one.
Repeatedly hurled from his hand, the dart repeatedly came back to him.
Even such was the might and glory of the adorable son of Agni. With
redoubled heroism, and energy and fame and success, the god pierced
the mountain and slew hundreds of daityas. The adorable god, having
thus slain the enemies of the celestials, was worshipped and honoured
by the latter and obtained great joy.
This is how BORI edition (Chapter 45 in BORI critical edition) of Mahabharata describes this event. Bibek Debroy translates it as follows:
The kinnaras were anxious, because they
were struck by the descent of the javelin. Hundreds and thousands of
daityas were crushed. They emerged from that blazing mountain, attired
in excellent and colourful ornaments and garlands. Kumara’s followers
proved to be superior and killed them in the battle. Pavaka’s son (Agni's son i.e Kumara), the
destroyer of enemy heroes, shattered Krouncha with the javelin. The
great-souled one divided himself into one, and also many forms. In the
encounter, he repeatedly hurled the javelin from his hands. Thus, did
Pavaka’s son repeatedly show his powers. He shattered Krouncha and
killed hundreds of daityas.