Aluminum project box: should BNC conenctors touch it?
Ground them all!!!

In the large majority of cases the BNC connectors should make good electrical contact with the box and the whole assembly should be at ground potential.

There are specialist cases where this is not so but by the time you encounter them you will know why you may wish to do it - and will sit agonising over whether you in fact should and at which end and more. 

The material below relates to why sometimes you shouldn't or what happens when you do or what issues relate. BUT for a start, ground them all, and read up on the why not's for future reference. 



For later years:

Ground loops in testing:

Excellent discussion




A ground loop results when two or more separate ground paths are tied together at two or more points. The result is a loop of conductor. Connecting the ground lead of an oscilloscope probe to the ground in the circuit-under-test results in a ground loop if the circuit is “grounded” to earth ground. Typically the metal chassis of both scope and device under test are connected to safety ground and internal power supply common. Scope probe ground connects to scope chassis at the input BNC connector.

In the presence of a varying magnetic field, this loop becomes the secondary of a transformer which is essentially a shorted turn. The magnetic field which excites the transformer can be created by any conductor in the vicinity which is carrying AC or changing current. The potential difference seen on oscilloscope probe ground can range from microvolts to as high as hundreds of millivolts.




Mysterious ground
Related enough to need to be aware of.



Video ground loop breakers - Expert anti-agonising devices.

Ground loop isolator  

SP-G01 CCTV Video Ground Loop Isolator, Coaxial Cable, BNC





And another



Good intro discussion


What is a Ground Loop

A ground loop occurs when there is a potential difference between inter-connecting equipment such as the camera and DVR and is due to an imbalance in the power loading of the two systems.  The potential difference cause a loop current to flow between the two inter-connecting equipment and as a result any interfering source such as 50 Hz mains power is picked up by the ground current flowing through the braiding of the coaxial cable instead of it acting as a shield to protect the interfering source from entering the coaxial cable.  

The 50 Hz mains power or hum is then passed on along with the wanted video signal from the camera and displayed on the monitor as a rolling hum bar