Grounding an Enclosure
Technically, any metal chassis device powered by an AC outlet needs to be Earth grounded. By itself it does not meet the 'double insulated' criteria, as it is metal.

However you do have options. If the wall-wart is a 3 pin device then it is Earth grounded so it can pass the ground to the (-) side out the DC output. If you connect this to your chassis then it is Earth grounded.

If you have a 2 pin wall-wart then it is isolated from the AC lines, but capacitive leakage can induce 10's to 100's of micro-amps of current into the metal chassis.

In which case your options are, in order of preference:


Buy a 3 pin grounded wall-wart.
Buy a plastic enclosure and have as little metal exposed as possible. You will notice the many 2 pin wall warts power devices with plastic enclosures, while large metal equipment may use a grounded desktop supply.
Add a green ground wire (14 to 18 gauge will do) from the chassis to a ground screw at the AC outlet, usually holding the cover plate on.


If you have a 2 pin wall wart you can check leakage current with a DVM set to read milli-amps. Measure from both + and - DC outputs to the Earth ground of the AC outlet. If less than 100 uA you are ok. If close to or over 1 mA then you have a shock hazard. Do not measure volts to ground as the meter has almost no load compared to the human body and would show only floating 'noise'.