Is port mapping or port opening safe?

There's no such thing as port opening.
The proper term is called port forwarding.

If you want to setup a server, then you would use port forwarding.

Port forwarding is not safe.
If you open a port, then you give people permission to access a certain
part of your computer, and maybe it can be exploited.

You would have to know what you are doing.
If you don't what you are doing, then you might have problems.

Port forwarding is a common practice and is ultimately required for hosting and sometimes just to play online. Just forward the ports that are required and everything will be fine. I host 24 hour servers out of my house for a number of games and I have no problems.

You may not know it, but you already have ports open to activity on the internet such as port 80, which is used to access the world wide web. Just don't open every port in existence. If someone wants to hack you, they will use a common port, not some random port that they don't know is open or not.