Q:

What does "has done half" mean?

"Did pretty well out West, didn't you?" asked the policeman.
"You bet! I hope Jimmy has done half as well. He was a kind of
  plodder, though, good fellow as he was. I've had to compete with some
  of the sharpest wits going to get my pile. A man gets in a groove in
  New York. It takes the West to put a razor-edge on him."

This is from the short story "After twenty years" of O.Henry. 

A:

The phrase here is "half as well", three words together functioning as an adverbial modifying has done. So the unabridged response from that person should read "I hope Jimmy has done half as well as I did." The last part "as I did" is omitted in the sentence.
The "as... as" structure is used in English to show equality in comparison. So "half as... as" indicates that in this comparison one thing roughly equals half of the other. For example:

I am jealous of how your  party was able to attract so many people. Half as many showed up at our party.

So the speaker is saying the number of people who showed up at their party is only half the number of people who showed up at the other party.
In your example of course the comparison is more figurative than literal. The policeman comments on the speaker's financial success out West (in the American West). And that person replies "I hope Jimmy is at least somewhat successful too."
† My translation for the phrase at issue into Japanese would be 半分の成果

