The short answer is no. The long answer is -- wait for it -- it depends. Sorry to have to give such a useless answer.

When investors or competitors are looking at your company and deciding whether they want part or all of what you have, they are going to put together a comprehensive analysis of the value of your company. Meaning, they are going to try and identify each and every asset you hold (physical property, intellectual property, key employees, and company goodwill) and decide what your company is worth. Then they are going to put together an estimate of what your company will be worth in the future.

If your business model includes the use or production of open-source software, then the value of your intellectual property (software copyrights and patents specifically) will probably be lower than if your business was developing/selling proprietary software. Thus, if your business's biggest asset is the software you develop, then your companies value is probably lower by producing software that doesn't produce licensing fees. But potential investors are still going to look at the value of other assets such as whatever key employees you might have on your team. Currently, most business's greatest single asset is their brand (Google is currently biggest with some $150 billion valuation).

To address your hypothetical, if Dropbox's client were open source then anyone could put together a client of their own to use with Dropbox. We'll assume that Dropbox would still produce their own client and charge the same amount for each tier of service. So the question then becomes, does the existence of alternative clients devalue the Dropbox brand, reduce the number of paying users, or allow competitors to poach users that would otherwise choose Dropbox?

In the case of Dropbox I think we can safely assume that the answer to all three questions is an emphatic 'no'. First, Dropbox has plenty of competitors that offer nearly identical products that they developed without the help of the Dropbox client: Google Drive, Apple iCloud, and Microsoft Office 365 to name a few of the bigger ones. An open source client wouldn't have had any effect on the competitiveness of the market. Second, the number of paying users would probably remain the same whether they were using a proprietary client or an open source client, because Dropbox is really charging for a service rather than software license.

Finally, we look at the effect on the Dropbox brand. It is possible, though I would argue unlikely, that if Dropbox allowed developers to freely put together their own clients for use with the Dropbox service that those unlicensed clones would diminish the Dropbox brand value. I think this is an unlikely result because I think that the vast majority of users would stick with the official Dropbox client and ignore the clones. Geeky power users might want to play around with customizable alternatives but they are a tiny niche market. Like most successful products, Dropbox's revenue is driven by average, non-tech-savvy people who just want a simple product that works.

Dropbox's potential is driven by the companies recognizable name, the product's ease of use, and the reputation that the company has built as being reliable and trustworthy. The product itself was never very innovative.

Companies who are developing highly specialized software, or patent-heavy products stand to lose a lot more value by going open source.

If you want to give some specifics about your company I would be more than happy to discuss how you might avoid losing value by going open source.