First and foremost, it should try to alleviate the cause, by doing things like stopping any bleeding, and increasing red blood cell production. But let's say these things aren't sufficient, or aren't done quickly enough. What compensatory measures can be put in place?
Well one thing that can be done is to shunt blood to the tissues that most need it. This means that the important organs get more blood than usual, even if the haemoglobin concentration (and thus oxygen content) of this blood is decreased. Indeed, the body does make use of this trick, and that's one reason why the sclerae of an anaemic patient are so pale. It isn't just that there is a decreased (red) haemoglobin concentration; it's also that there is simply less blood being pumped there, as more is being diverted to other areas.
Another compensatory mechanism is to increase the cardiac output, principally by increasing the heart rate. In this way, more oxygen per second is transported to the tissues, which helps to make up for there being less oxygen per unit of volume in the blood.
Does that make sense? Perhaps an analogy will help. Think of fish swimming down a river, with you having to catch a certain amount of fish per hour. A problem strikes: this year has been a poor one for the fish, and there are fewer than usual fish per kilolitre of water. What can you do? The tachycardia option would be the equivalent of making the fish swim faster down the stream, so that you can keep your quota-per-hour to normal. (The shunting option would be represented by closing down certain rival streams to divert enough fish per hour towards you.)
Incidentally, this increased cardiac output often produces a soft ejection systolic murmur, as a result of the increased speed and volume of blood passing by the aortic valve.
(Oh, what other one thing could you do? You could try to make sure that haemoglobin gave up its oxygen load more easily - by doing any of these things, thereby "shifting the curve to the right".)