Learn Finnish: Älä anna selaimen tallentaa tarpeettomia evästeitä. - Do not let the browser save unnecessary cookies.

This is the core syntax for verbs of permission like antaa (to let) and sallia (to allow). When you say 'let [something] do [something]', the entity being permitted (here, the browser, selain) acts as the subject of the following verb. This 'infinitive subject' must be in the genitive case (selaimen), followed by the first infinitive of the action (tallentaa).

Would selaimen change its case if this wasn't a negative command?

No. The subject of the infinitive (the one given permission) always stays in the genitive case with antaa, regardless of whether the main verb is positive, negative, imperative, or a regular statement.

Why are tarpeettomia evästeitä in the partitive case? Is it just because of the negative command Älä anna?

The negative command definitely forces the object into the partitive, but even in a positive sentence ('Let the browser save unnecessary cookies'), you would likely still use the partitive plural here. This is because 'cookies' represents an unspecified amount or an ongoing, open-ended action of saving them over time, not a specific, closed set of cookies.

How do we break down the word tarpeettomia?