The licensing fee you paid to Delaware is a deductible business expense provided it is reasonable and necessary for your business. It appears that you registered or incorporated a business in Delaware, but all business activities were performed in California. While it is not clear from your questions whether your business is a Limited Liability Company or a Corporation, because you indicated that no business activity, nor any resulting income, was conducted/earned in Delaware, it would appear there is no need to file a Delaware state return.
Even though your business generated no income, if you want to claim your Delaware licensing fees as a business expense, such a deduction is typically claimed on a Schedule C (appropriate for single member LLCs or sole proprietors). Business expenses of this type can be entered into TurboTax Deluxe Desktop, thus it appears you have the correct version of TurboTax. Although, if you registered a corporation in Delaware, TurboTax Deluxe Desktop will not generate a corporate return. Additionally, given that your business generated no income, any reasonable and necessary business expenses you include on your return will likely generate a loss for the business.
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