There's no one word that means the exact opposite of 'to worry', but there are many ways you can describe comforting someone. Here are some examples:
To quell someone's fears. 'I hope my assurances will quell your fears.'
To ease someone's mind. 'If it will ease your mind, we can call today and ask them.'
To reassure someone. 'I had hoped my story would reassure you.'
To calm someone. 'You need to calm down." ('calm' is almost always followed by 'down')
To console someone (if they are already very upset). 'I had hoped my visit would console you some.'
If you want to say that the person never got upset in the first place, you can use:
To be worried. 'I didn't want you to worry about it, so I didn't tell you.'
To fortify someone. 'Rest and relaxation will help fortify you before the busy season.'