Can you share more detail about what your customers really want when they ask if your product can be more open source? There are various open source strategies and depending on what your customers are asking for some may be mutually beneficial.
Open code, Restrictive license:
It is possible to share source code while keeping that code licensed under a commercial license so that customers can customize your product. Atlassian does this for some products. Epic Games provides full source code access for Unreal Engine. In both cases, customers must first buy the software and are not allowed to copy it. Clearly, it helps to have some money and lawyers to enforce such agreements.
Many web companies release free, open source code libraries that make it easier for developers of other software to build custom applications on those APIs.
Basecamp is the best (and maybe a unique) example of a company that extracted a generic framework from its commercial applications and distributed that framework as open source (Ruby on Rails).
Facebook, Twitter, Google, Apple, and others open source tools that help run their businesses or help to make software, without releasing the specific code they make money off of.
Open product, Commercial service:
Wordpress is the best (and maybe a unique) example of an open product that is supported by a paid service business. As with Wordpress, hosting of complex or mission critical installations is a typical service offering. Wordpress also demonstrates two additional ways to open up a product: themes and plugins. Themes allow artists to customize software at a UI layer, decreasing work for the core team. Plugins allow developers to extend functionality. You'll find a mix of open source, closed, free and commercial themes and plugins.
Warrantee, Support, Upgrades:
Red Hat resells open source Linux, adding a warrantee and support contract and consulting services. You'd be surprised how many businesses there are that simply won't use open source without buying it from a third party. Some companies will choose an inferior product because there is a customer support representative they can call. So, while you may lose revenue from small businesses or hobby projects it is possible to retain revenue from large corporations that have a lot to lose financially.