This looks pretty straight forward, actually:
RewriteEngine on RewriteRule ^/?api/dev/v1/(.*)$ /api/v1/$1 [END]
In case you receive an internal server error (http status 500) using the rule above then chances are that you operate a very old version of the apache http server. You will see a definite hint to an unsupported [END] flag in your http servers error log file in that case. You can either try to upgrade or use the older [L] flag, it probably will work the same in this situation, though that depends a bit on your setup.
This implementation will work likewise in the http servers host configuration or inside a dynamic configuration file (".htaccess" file). Obviously the rewriting module needs to be loaded inside the http server and enabled in the http host. In case you use a dynamic configuration file you need to take care that it's interpretation is enabled at all in the host configuration and that it is located in the host's DOCUMENT_ROOT folder.
And a general remark: you should always prefer to place such rules in the http servers host configuration instead of using dynamic configuration files (".htaccess"). Those dynamic configuration files add complexity, are often a cause of unexpected behavior, hard to debug and they really slow down the http server. They are only provided as a last option for situations where you do not have access to the real http servers host configuration (read: really cheap service providers) or for applications insisting on writing their own rules (which is an obvious security nightmare).
For this to work the rewriting module needs to be loaded into the http server, obviously. In addition, if you decide to use a dynamic configuration file (".htaccess" style file) despite what I explained above, then you need to enable the interpretation of such file for the requested location (look for AllowOverride in the apache documentation). The file needs to be placed in the DOCUMENT_ROOT folder of the http host responding to the request and it needs to be readable by the http server process. And of course it needs to have valid syntax. If all that is given and things still do not work, then you definitely should take a look into your http server's error log file. That is where the engine will log all issues it runs into. You need to monitor that file anyway when working on implementations in a web environment, no way around that.