Which device is responsible for translating the private IPv4 address to a public IPv4 address when using static NAT?

Router which will show the device that responsible to translating the private ipv4 address to a public ipv4 address of using the static NAT.

Router is the device capable of doing NAT settings.

Router* if you did not have any firwall-appliance on backend 

 Router and Firewall are responsible for NAT

A Firewall or a router could be used , usually these are public facing devices

Router OR firewall devices 

the Device which is resposible for doing do is Router 

Router is the device which is responsible fot translating IPV4 private IP to Public IPV4 IP address.