A high beta means that the stock is more volitle - it has a higher probability of dramatically going up or down in value. The more volitile a stock is, the more intrisic value an option has.
Option premiums are made up of two items - intrisic value and time value. Premium = Intrisic + Time. The further off the expiration is, the more time value the option has. The higher the intrisic (volitility of underlying stock, current price of underlying stock, ect), the higher the premium.
So yes, a higher beta would mean it is more volitile, and more volitile underlying securities would likely have higher intrinsic value creating higher priced put options.