Is it better to close an internet cafe where lawful and unlawful things are mixed, despite striving to monitor it as much as possible, or to keep it open to prevent the spread of vice in other, less observant places?

The internet is a double-edged sword; it is a useful tool for da'wah (preaching Islam), but it can also be a cause for spreading immorality and discord.

Therefore, if an internet cafe can be managed in a way that pleases Allah, then opening it is preferable. Otherwise, avoiding it is better, because nothing equals safety in one's religion.