A person who gets on a bus is a 'rider' or 'passenger'.

It's definitely not border. That means 'edge'. You could say bus boarder, and that would probably be understood, but it is not a common or standard phrase. It would be better to just say 'the person that gets on a bus', if you want to emphasise the fact of them physically getting on. Otherwise, as Joe said - the normal way to describe them is as a passenger or a rider.

Once you are on the bus, you are a passenger. However, if you want to specifically refer to passengers who are in the process of getting on the bus, you could use 'boarding passengers'. The opposite is 'disembarking passengers', or, sometimes, 'alighting passengers'. There could be a sign which says "Passengers who are boarding or disembarking are advised to exercise caution". Of course this is just a fancy way of saying - be careful when you get on and off the bus!

UK English. The people on a bus are always passengers. When you get on a bus you Board the bus. you ride on the bus.

Well said, Hilary. We all spoke around it, but you answered the question best.