You could write 'he or she' and 'his or her' every time, but it gets a bit tedious. The other option, and the one that I prefer is to use 'they' in place of 'he or she' and 'their' instead of 'his or her'.
There is an important difference though, compare these two:
He or she does not have money.
They do not have money.
So you are using "They" as if you were talking about multiple people, but in English it is understood you are talking about only one person. Sometimes it could depend on context. If I didn't already know you were talking about a single person, it might not be clear for me.
But in regular conversation it is very common:
"Look there! Is that a person sailing in this bad weather? They do not stand a chance!"