Oh, there are MANY games I can criticize for THIS nonsense!
...At least four. Fable 2, Fallout 3, Fallout New Vegas, and Skyrim.
Fable 2 I can criticize the hardest, however, because if memory serves, it allows you to give variable amounts of gold like this all at once, and it never changes anything for them. It's just a way to turn gold into "good" points.
Fallout 3 gets the least criticism by virtue of what you give being purified water, which naturally wouldn't change their situation in a major, long term way. Though, now that I've stopped to think about it, why "give them some caps" (Fallout's currency) isn't an option is beyond me (aside from the obvious "a game has to draw the line on options somewhere" answer).
In New Vegas, there's at least one guy you can give caps to in exchange for information, and you can wind up giving him a fair amount if you're persistent, and... Wait. I'm pretty sure he actually does get a new hat at some point if you do it enough.
Okay, FINE, New Vegas. Just show off, why don't you.
Meanwhile, in The Elder Scrolls V: Skryim, there are people in every major city you can give gold to, but only one gold coin at a time regardless of how much you have (you temporarily get an improved barter skill if you do this, so it's absolutely worth doing from a game play perspective).
Skryim takes second place for my criticism, however, because at least one of those characters begging for coins is a skill trainer in pickpocket, a skill that's rather awkward to level without taking advantage of her tutelage. If you do get training from her, the amount of gold you wind up giving her could be more than enough to acquire any home in the game. Granted, it's canon she'd need permission from someone in power to acquire one within the city, but you get my point.
BUUUUUT I also have to PRAISE Skyrim, because the ability to adopt younger beggars became an option with one of the DLCs, so it is actually possible to make a huge difference for some characters.