The dishonor spoken of here is not about popularity or reputation.
"Dishonor is worse than death" is not a general statement that encourages people to disregard life or sacrifice their lives for the sake of reputation. It describes a specific situation where Arjuna, who has long upheld courage and duty, suddenly withdraws from righteous battle.
Here, the disgrace refers to damaging one's own way of life and responsibilities, not through popularity votes from others. For someone who values honor, it is a pain that deeply wounds the sincerity at the core of their being, more than any external evaluation.
Japanese Translation from the Geeta Press Edition - Chapter 2, Section 34: "For a man of honor, dishonor is heavier than death."
Sanskrit original text
अकीर्तिं चापि भूतानि कथयिष्यन्ति तेऽव्ययाम् ।
सम्भावितस्य चाकीर्तिर्मरणादतिरिच्यते ॥ ३४ ॥Romanization
akīrtiṃ cāpi bhūtāni kathayiṣyanti te 'vyayām |
sambhāvitasya cākīrtir maraṇād atiricyate || 34 ||Furthermore, people will pour endless disgrace upon you. And the disgrace that is brought to someone who is respected by others is worse than death. (34)
In the sentence, "People will further pour endless disgrace upon you," the adverb "further" is used.
This indicates that retreating from battle not only fails to fulfill Arjuna's duty but also diminishes his reputation and exposes him to blame.
The beings in heaven, the sages, and all people on earth will speak ill of him in various ways.
Furthermore, this dishonor is not a minor one, nor will it end in a short period; it will be permanent. Therefore, the idea of withdrawing from battle was completely unacceptable to him.
The master is trying to convey the following through the saying, "Dishonor brought upon a person who is respected is worse than death."
If Arjuna concludes that he loses nothing by public dishonor, then that is not a correct conclusion for him.
For someone who holds a wide reputation and is loved and respected by people, losing trust and honor is more painful than death. Therefore, when faced with such dishonor, he would not be able to accept it. Because he is known throughout the world as a hero and a great person, and his fame extends far and wide, from heaven to the underworld.