For some bizarre reason, one of my cousins believes I've hacked into her computer though Outlook Express and that I've deleted hundreds of her emails and stolen personal information from her by that means. She honestly believes this because several weeks ago, I make an attempt, over the phone, to help her set up Outlook Express on her computer. She's barely computer literate, so she could never get it to work. A few days later, she starts accusing me of setting it up so I could hack into her computer that way.

I wouldn't know how to hack into a computer if you handed me an ax, and from everything I've looked at online, it's not possible to gain remote access to someone's computer through Outlook Express. I would have to be physically sitting at her computer.

I don't even live in the same state she does, so it's not as if I have direct access to her computer, but she continues to accuse me of "going into her computer" through Outlook Express.

She has been sending me emails on a daily basis for more than a month, threatening to have me arrested for doing who knows what to her.

How can I convince her that it's probably not even possible to do what she's accusing me of doing?

In all honesty, it's probably best to simply ignore her and let her work her way through this personal mess.
You could spend hours explaining why it is not possible for you to do what she claims, but she still is unlikely to believe you.
As for law enforcement, without any actual evidence nothing will happen in that regard.

Leave her to cool off for a while and stop responding to these ludicrous accusations.

I'm trying to ignore her. She continues to send me at least one email message a day, saying she can tell I'm "in her computer" because it's making odd noises. Yeah -- she's nuts.

So file the e-mails and don't response.
Simple enough ;)

Either that or tell her you have enabled SMOKE and that she needs to turn off and unplug her machine to stop any further problems.

I agree.

Just tell her once and for all that you didn't do it, it can't be done, and you will not be talking about this any further. Explain that any emails related to this will be deleted from this point on. Ignore her after this. Don't worry about law enforcement... I can just imagine how her phone call to them would go. Lol