If you handle credit card information there is the PCI-DSS standard (Payment Card Industry Data Security Standard, often just called PCI). It specifies a dozen requirements that all companies with payment card data must meet. It is quite controversial (see the linked Wikipedia article).
ISO/IEC 27001 is another certification that is meant to make security practices auditable and certifiable.
Although it's definitely a good idea to see what you can learn from standards like these, just like with HTML validation, there is in my opinion little value to your site's visitors. The standards specify how an organization must do things, but are easily circumvented if the organization chooses to. In other words, they're nice on paper. Most customers won't know what they are and what it means if you put an "I'm certified" logo on your site.
Your best bet is to actually do things properly, and build an image of reliability and trust. Word of mouth spreads fast on the web, both if you do things well and if you do things badly.