Adding players to a game can be a fun and great experience for the game. It can add a new dynamic to the game in general, and can bring new energy to the players. The introduction of a new PC can test, push, and develop relationships. Not to mention the added skillsets to the group, and new avenues for challenges. That being said, there are some steps I've found it is best to take in order to keep your game safe and fun. After all, while a new player can bring a lot of good things...bad chemistry can also make it a bad fit which can damage the game.
Step 1: Talk To Your Players About Adding Someone
The first thing to do is talk to your players about adding someone. Do t hey even want someone else in the game? It could be they're happy with the game the size it is. Less players means more GM time for each individual after all, and if the group isn't having problems overcoming encounters, they may be happy that way.
You should also talk about who the group would be interested in inviting. Is there someone who has asked to join? Do you have someone in mind? Does another player have someone in mind? For that matter, do you want to bring someone new you haven't gamed with before, or would you rather someone you know and have played with?
Step 2: Is Everyone Ok With This Person?
Everyone should be ok with the new player. And by that I don't mean just ask the table if everyone is cool with it, but make sure people are in private. To me, one player saying no means we don't go with that person. The player in the game is there first, and if they're not going to be comfortable playing with that person that is all I need. I don't ask questions, I don't need to know.
If you need to know, talk to them, but just keep in mind it is a question of comfort. Sometimes a player is just reaching beyond their normal boundaries with a character and won't feel safe to do that with certain people. Or there could be bad blood you don't know about. The idea is to preserve the game and make it better, not make someone uncomfortable with a new person.
Step 3: Ask The Player
With everyone on board, you can make the offer to the player. Be sure to talk to them about the kind of game it is, what the expectations are for players, and what the tone/theme/etc is to be for the game in general. It can also help to give a rundown of what is already in the game, what could work well coming in, and what things are maybe best to avoid.
Step 4: Review the New PC Carefully
If the player accepts and makes a character, review it carefully. Remember, this isn't a character walking in with four to five other new characters to form a group. This is someone coming in to an existing group. If you see red flags that would make the group not take them in, point that out and try to correct them.
For example, if someone in the group **hates** goblins and kills them on sight, a goblin PC is probably a bad idea. It is an even worse idea if you're playing more default D&D where anyone can kill a goblin and be sure they are doing the right thing. A lone Lion in a Crane game is also likely going to be awkward. As would a social character in a murder hobo game.
Step 5: Work With the Player to Intro the Character
Start the new character off with a reason to join the group. Work with them and their story to make it make sense. And when the session happens, help steer things towards it. Once the person is with the party, they can prove their merits to the party on their own.
A Trial Period Works Best
Let everyone know you're starting with a trial period. Try to bring the character in with something that will work as a short adventure (1-3 sessions.) This gives everyone a chance to see if they work well together - including the new player to see if they like the group. After, if everyone is ok, just continue as normal.
A Final Note
This may seem like a long process, but in my experience it works well for smooth transitions of new players into games. I've never had a problem with someone coming in on a trial basis - and if they balk at it, it's probably a sign they aren't necessarily a good fit. After the trial period the person is a player in the game, like everyone else. So it is good to keep up with your players privately to make sure no one is holding back for societal norms.