There are ethical and compassionate reasons why you might or might not give money to people who beg on the street. The issue of homelessness and begging are related but are not necessarily the same. Many of those who beg are not homeless and the majority of homeless people do not beg.
John Kurt (Chief Executive at West London Mission) works with people affected by homelessness and addiction and puts forward his reasons for not giving money to people begging on the street in this article.