Praise be to Allah.
It is not permissible to work or invest in internet cafes unless they are free of evils, which includes not allowing the visitors to go to haraam websites, by denying them access to such sites, or by throwing them out if they insist on using them. That is because Allaah says (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment”
“Those among the Children of Israel who disbelieved were cursed by the tongue of Dawood (David) and ‘Eesa (Jesus), son of Maryam (Mary). That was because they disobeyed (Allaah and the Messengers) and were ever transgressing beyond bounds.
They used not to forbid one another from Al-Munkar (wrong, evildoing, sins, polytheism, disbelief) which they committed. Vile indeed was what they used to do”
[al-Maa'idah 5:78, 79]
And the Prophet (peace and blessings of Allaah be upon him) said: “Whoever calls others to right guidance will have a reward like that of those who follow him, without that detracting from their reward in the slightest, and whoever calls others to misguidance will have a burden of sin like that of those who follow him without that detracting from their sins in the slightest.” Narrated by Muslim, 2674.
And he said: “Whoever among you sees an evil action, let him change it with his hand (by taking action); if he cannot then with his tongue (by speaking out); and if he cannot then with his heart (by hating it and feeling that it is wrong), but this is the weakest of faith.” Narrated by Muslim, 49.
Shaykh Ibn Baaz (may Allaah have mercy on him) said: “Denouncing in one's heart is obligatory upon everyone, which means hating and despising the evil and separating from those who do it if one is unable to change it with one’s hand or tongue, because Allaah says (interpretation of the meaning):
‘And when you (Muhammad) see those who engage in a false conversation about Our Verses (of the Qur’aan) by mocking at them, stay away from them till they turn to another topic. And if Shaytaan (Satan) causes you to forget, then after the remembrance sit not you in the company of those people who are the Zaalimoon (polytheists and wrongdoers)’
From al-Durar al-Saniyyah fi’l-Ajwibah al-Najdiyyah, 16/142.
If it is not possible to control this café and prevent evil from taking place therein, then it is not permissible to open such a place, so as to avoid falling into sin.
If you give up this work seeking to draw closer to Allaah and to keep away from dubious matters, then receive the glad tidings of the hadeeth of the Prophet (peace and blessings of Allaah be upon him): “Whoever gives up something for the sake of Allaah, He will compensate him with something better than that.”
We ask Allaah to give you strength and to bless you, and to compensate you with something better.
And Allaah knows best.