How would you respond to a drunk non-Muslim person when he is insulting (i.e. usually in the streets)?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The appropriate and correct solution towards whoever insults you, swears at you or behaves in a foolish way towards you, is to avoid him whether he is a Muslim or non-Muslim.

Allah Says (interpretation of meaning) about His believing slaves: {…and when the foolish address them (with bad words) they reply back with mild words of gentleness.}[25:63].

Allah also Said to His Prophet (Sallallahu Alaihi wa Sallam): {Show forgiveness, enjoin what is good, and turn away from the foolish (i.e. don't punish them).}[7:199].

The Prophet (Sallallahu Alaihi wa Sallam) his companions and the righteous people who came after them acted in accordance with this noble standard.

It is reported in Saheeh Bukhari that Ibn Abbas (Radiya Allahu Anhu and with his father) said: 'Uyaynah bin Hasan bin Hudhaifa came to his nephew Al-Hur Ibn Qays, and he was among the people with whom Umar dealt closely, the reciters of the Qur'an were the people who used to sit in the gathering of Umar (Radiya Allahu Anhu), and used to consult them whether they were elderly people or young.  So Uyaynah told his nephew: ‘Does the Caliph have any favour on you so that you may intercede for me and ask his permission to let me see him’.  He replied: ‘I will seek his permission to let you see him’.  Ibn Abbas said: ‘Al-Hur asked the permission of Umar to let Uyaynah see him.  So Umar granted the permission.  When Uyaynah entered to Umar he said: ‘It is a word of reprimand – Oh Bin Khatab, Oh by Allah you do not give us abundantly, and you do not judge between us with justice.  Umar became angry and was about to hit him, so Al-Hur told him, ‘Oh, Amir Al-Mu’minin (Leader of the Muslims) Allah Told His Prophet (Sallallahu Alaihi wa Sallam) {Show forgiveness, enjoin what is good, and turn away from the foolish (i.e. don't punish them)} and this person is among the foolish ones.

I swear by Allah, Umar did not go beyond the verse, and he was always acting upon the Book of Allah”. He acted according to the verse.

So, dear brother, look to this noble character and great manners.  Unfortunately a lot of people today are behaving contrarily to this.  They respond to foolishness with foolishness, or even worse.  The Prophet (Sallallahu Alaihi wa Sallam) said: "If a person insults you with what he knows about you, then do not insult him with what you know about him; he is the one who is sinning". [Reported by Abu Dawud, Tirmizi and Ahmad] So try to imitate these noble people and turn away from foolish people, your status will increase, and your position in the society will be greater, and you will be respected with dignity.

Allah knows best.