Options for fixing outside cement staircase?
Here are some options:


Keep a broom nearby and sweep away excess water.  While this does not actualy fix anything, it will eliminate the excess water.
Create a drain.  Drill a hole at the lowest spot of each step for the water to escape to.  This solution is not optimal and may only alleviate the problem.  One will need a masonry bit and hammer drill to get through the step.   Also consider the effects of having water being drained under the step, which also may not be optimal.
Level the steps.  There are some low spots, so another layer of concrete on top should allow the water to escape to a drain below the steps.  However, getting a consistent and durable finish across all the steps will be a challenge.
Refinish the steps. Add a new finish on top of the concrete and in the process eliminate the low spots so the water does not pool on each step.  As a plus the drab lichen colored gray would be updated.
Rip out the entire thing and replace/redo with a better step/drainage solution.


I'd probably go with step 4 provided the settling has stopped.  If the foundation is no good or drainage is not adequate, then option 5 would probably be the best.