O'Henry was a short story writer and often used language in poetical ways. For many English speakers, 'move up' suggests becoming better or more important. For instance, if I get a much better job, you could say I've 'moved up in the world.'
In this sentence, O'Henry uses the words 'moved up' to enrich the image of the policemen being 'impressive.'
Yes, the words 'moved down the street' are technically correct but in the context of O'Henry's sentence, 'moved up the street' is better.