Correct spelling, explanation: the word worsened is a derivative of the infinitive to worsen. The past simple/past participle form must always go along with its root. That’s why you can’t spell it as worstened. This verb’s conjugation is regular, so it’s created by combining two elements: the basis and the suffix. Therefore, worsened is the correct version.

Definition of worsened:
verb (past simple/past participle), to become worse, or to make something become worse
The country’s crisis only worsened.
My marital relationship has definitely worsened recently.

Collocations with worsened:
Some most commonly used collocations include: worsen because of and worsen due to.
My hair’s condition has worsened because of the low quality of the water in this city.
Kate’s school grades got worsened due to her parent’s divorce.

Incorrect spelling, explanation: you can’t spell this word as worstened because this version doesn’t exist in any dictionary. This error may come from the wrong assumption that this verb is based on a superlative adjective – worst – when in fact it’s based on a comparative adjective – worse. It’s really important to remember that. The only correct form is worsened.