I use 1Password and love it. I can tell after using other products that they have put a tremendous attention to detail into it. It is really easy to use and thoughtful in its design compared to the alternatives.
It just makes no sense to centralize your most important secrets into a single attack point for which you have no idea about its inner workings.
Look at Heartbleed and how lack of funding led to a horrible bug being missed. There has been other open-source projects hit by similar issues. Just as there are closed-source projects being hit with their own issues.
The nature of the code license does not, and I stress this strongly, lead to anything being more secure than other solutions.
A properly funded and talented team of developers working on an open source project is just as secure as a properly funded and talented team of developers working on a closed source project.