If you write a put option, it has downside limited only by the underlying zero price of the security minus the premium. If you write a call option, your downside is theoretically unlimited. If you buy a put or a call your downside is limited to the option premium.
edit: i should say writing a put option (short put) your potential downside is the difference between the strike price of the option and the premium, if the underlying security goes to zero.
If the underlying security goes below zero, that doesn't increase your potential downside. The option you sold will force you to buy the security, but it won't force you to sell it.
edit: i should say writing a put option (short put) your potential downside is the difference between the strike price of the option and the premium, if the underlying security goes to zero.