Many years ago I was homeless for a few months and I never asked anyone for money, I didn't even consider it. I met a couple of people in similar circumstances who shared the same attitude.
I did see, however, many other 'homeless' people who spent their entire day asking people for money. This was in a fairly small city where you soon start to recognise the same people over and over agin.
For the most part, these people were actually not homeless in the true sense of the word, they were staying in one of many hostels where you get fed, access to a bathroom/shower and a bed for the night. They were people with drug/alcohol probelems who were being enabled by a sysytem initially created to bring about an end to homlessness yet has evolved into one that perpetuates it - because if you are just someone who's made a few bad decions in your life then you will get no help from them at all.
What I would say to you is that if it makes you feel better giving people money then by all means do so. The recipient will also feel better for a while but they'll probably just be digging themselves further down the same hole. In the end, you're making no difference.
I was lucky, I got help from a couple of random strangers who'd noticed me walking around aimlessly. I got back on my feet fairly easilly after that.
Some are not so lucky and those most deserving of a bit of help are most likley the people you never see. And the reason you don't see them is because the genuinely homeless people who can be easily helped are the least likley to ask for it - that's why they ended up where they are.