The simple answer is that, at least when it comes to exchange traded options, an option can't have a negative strike price .
Remember that an option gives the holder the right, but not the obligation, to buy or sell an underlying security at a set price (strike price) before a set date in time. If the strike price were to be negative, it would mean that it would cost you a negative amount to buy or sell a security. If it was a call option on a stock with a strike price of $5, it would mean that if you exercised the option you would receive $5 for each share you bought. This would mean that no matter what happened to the price of the underlying security, the option holder would exercise the option.
The strike price of an option should be related to the price of its underlying security. In most cases, the price of these securities can never be negative, so there is no reason to have an option with a negative strike price.
However, this doesn't necessarily mean that you couldn't have an option with a negative strike price. The reason for this is that an option is simply a contract between two parties. A contract is completely customizable and could even have an option with a negative strike price.
For more insight, see the Options Basics Tutorial.

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Strike Price
The price at which a specific derivative contract can be exercised. ... 
In The Money
1. For a call option, when the option's strike price is below ... 
Near The Money
An options contract where the strike price is close to the current ... 
Average Strike Option
A type of Asian option in which the strike price is based on ... 
Bear Call Spread
A type of options strategy used when a decline in the price of ... 
Strike Width
The difference between the strike price of an option and the ...