"The internet doesn't work" is the correct answer. Think about how the verb is conjugated..

I don't work

You don't work

He/she/it/the internet/my car/the television doesn't work

We don't work

They don't work

You don't work.

When we are discussing a singular thing, we use "doesn't".

My phone doesn't charge.

The computer doesn't work well.

The stove doesn't heat properly.

For plural things, or when we use "I"  or "you" we use "don't".

The cars don't look good

You don't like ice cream

I don't exercise enough.

I hope this is helpful to you! :)

The internet doesn't (does not) work.

2) Internet doesn't work