The resistance to the Bill of Rights was based on this very notion, the fear that adding an enumeration of some rights would eventually be seen as an enumeration of all rights and that any rights not explicitly enumerated would be considered as lesser rights not protected by the Constitution.
Of course, a right being clearly stated in the Constitution doesn't do much to protect it either. Consider all the limits on gun ownership that run counter to the second amendment. Even someone who is against guns should be of the view that a Constitutional amendment should be needed to allow for limits like those we currently have. For example, banning a mentally ill person from owning a gun or charging a licensing fee to own a gun. Both of these exists and are not considered to infringe on second amendment rights. Now imagine if the same logic was applied to other rights, such as a fee to vote or not allowing free speech to those deemed to have a mental illness (never mind the difficulty in determining what constitutes a mental illness).
Of course, a right being clearly stated in the Constitution doesn't do much to protect it either. Consider all the limits on gun ownership that run counter to the second amendment. Even someone who is against guns should be of the view that a Constitutional amendment should be needed to allow for limits like those we currently have. For example, banning a mentally ill person from owning a gun or charging a licensing fee to own a gun. Both of these exists and are not considered to infringe on second amendment rights. Now imagine if the same logic was applied to other rights, such as a fee to vote or not allowing free speech to those deemed to have a mental illness (never mind the difficulty in determining what constitutes a mental illness).