📜 Question: Saying "Bismillah" every time one raises the right hand to one's mouth to eat or drink, and saying "Alhamdulillah" (after every bite and sip) each time one places down the glass or utensil, is there any basis for this?
📝 Answer: No. The Sunnah when eating (and drinking) is to say "Bismillah" from the very beginning and then say "Alhamdulillah" at the end of one's drink or meal.
Question: What about when raising the glass or cup while taking three separate sips?
Answer: The person should say "Bismillah" (before the whole process of eating and drinking) and upon completion one must say "Alhamdulillah."
What has been reported is to say "Bismillah" at the very start of the meal and "Alhamdulillah" at its end. And if the person decides to repeat then Allah Subhaanahu wa Ta'aala may reward them with goodness (for saying additional Dhikr). However the actual Sunnah is that once is sufficient. You say "Bismillah" one time only before eating and drinking, then say "Alhamdulillah" one time after you're done with eating and drinking.
From the Fataawa of sheikh AbdulAziz Ibn Baz (May Allah have mercy on him. Ameen.)
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📖 Umar ibn Abi Salamah RadhiAllahu 'anhu narrated: The Messenger of Allah SallAllahu 'alaihi wa sallam said to me, "O young man, mention the Name of Allah, eat with your right hand, and eat what it is in front of you."
Source: Sahih Al-Bukhari 5061
Sahih Muslim 2022
Grade: Muttafaqun Alayhi (Authenticity agreed upon) according to Al-Bukhari and Muslim
📖 Hudhayfah RadhiAllahu 'anhu narrated: Allah's noble Messenger Muhammad SallAllahu 'alaihi wa sallam said, "Indeed, shaytan permits himself to eat the food over which the Name of Allah is not mentioned."
Source: Sahih Muslim 2017
Grade: Sahih (Authentic) according to Muslim
📖 Jabir ibn Abdullah RadhiAllahu 'anhu narrated: Allah's noble Messenger Muhammad SallAllahu 'alaihi wa sallam said, "When a man enters his home and remembers Allah as he enters and as he eats, then shaytan says to himself: 'you have no place to spend the night and no dinner.' When he enters his home without remembering Allah, then shaytan says: 'you have found a place to spend the night.' When he eats his meal without remembering Allah, then shaytan says: 'you have found a place to spend the night and eat dinner.'"
Source: Sahih Muslim 2018
Grade: Sahih (Authentic) according to Muslim
https://binbaz.org.sa/fatwas/23245/%D8%AD%D9%83%D9%85-%D8%AA%D9%83%D8%B1%D8%A7%D8%B1-%D8%A7%D9%84%D8%AA%D8%B3%D9%85%D9%8A%D8%A9-%D9%88%D8%A7%D9%84%D8%AD%D9%85%D8%AF-%D9%81%D9%8A-%D8%A7%D9%84%D8%B4%D8%B1%D8%A8