Painting the old utility stairs to the basement
In my opinion, that's the wrong approach. Cement and mortar don't do well in places that move, and stairs tend to move. 

I've restored several sets of old basement stairs in my homes, and new lumber is your best bet. I've gone with stained pine, which is beautiful but a bit soft. If you're going to paint, consider bullnose particle board. 

By rebuilding with 1x8 pine and your choice of tread you eliminate those gaps, remove any squeaks (by using construction adhesive on all joints), and leave a very nice base for some good floor paint or stain and urethane.