Best advice on Frayed, exposed wires on air purifier cord
It's hard to say how bad the wire is damaged but it could be dangerous so it's good you found out about it. Make sure you unplug it before wrapping electrical tape around the cord. 

Replacing the cord is a necessity. I replaced a cord on a Sharper Image air filter and I had to just about disassemble the entire unit to get to where the cord was attached to the unit. Unless you're very handy I wouldn't advise you doing this. Take it to an appliance repair shop or check with the manufacturer about having them fix it. I know this is a DIY site but sometimes we just have to bit the bullet and recommend getting a pro involved.