I've no idea what the usual fate of orphans was in classical Athens, though I'd wager that, like at most other times, it was an unhappy one. Hopefully someone else here can shed more light on that front.

At any rate, to my knowledge not too much is known of Socrates's children. There is, however, the following passage from Aristotle's Rhetoric Book II Chapter 15 that you might find interesting:

Being well-born, which means coming of a fine stock, must be distinguished from nobility, which means being true to the family nature -- a quality not usually found in the well-born, most of whom are poor creatures. In the generations of men as in the fruits of the earth, there is a varying yield; now and then, where the stock is good, exceptional men are produced for a while, and then decadence sets in. A clever stock will degenerate towards the insane type of character, like the descendants of Alcibiades or of the elder Dionysius; a steady stock towards the fatuous and torpid type, like the descendants of Cimon, Pericles, and Socrates

So, it seems that at least in the opinion of Aristotle, the children of Socrates did not really amount to anything. Aristotle, of course, would have been a position to know. Not only would his own lifespan have likely overlapped with Socrates's children, but of course Aristotle's beloved teacher was none other than Plato, who almost certainly would have known something of Socrates's children.