Yes, the power supply needs to be mounted in a suitable enclosure so no one can touch it.
Where did you buy it from? It looks dodgy. There's a double insulated symbol on it, then there's an earth terminal which is contradictory. Double insulated equipment doesn't need an earth connection and normally has a plastic enclosure and if it does have a metal box, there's a piece of plastic between it and the live parts.
It certainly needs to be earthed and I'd be tempted to earth the secondary too as it might not be designed with sufficient insulation.