Is regular "wood glue" suitable for lighter woods (e.g. balsa)?
Wood glues are fine for balsa.
While there are stronger glues, the added strength is usually unnecessary and strong enough should be sufficient. It is hard to generalize but &quot;white&quot; glues, like Elmers, are just PVA glues that are advertised as craft or general glues. There are also yellow PVA glues, which Elmers also makes (and the one I generally use for my indoor projects), that are geared more towards wood. However there is many similarities between them and in some cases the yellow is just colour added (FYI white vs. yellow is a large debate.).
A common adage among wood workers is that the glue joint is stronger than the wood itself and that is true. Also true is that balsa is one of the weakest hardwoods but it is ideal for a great deal of applications.  Go ahead and use either white/yellow glue for this. Some people would even use cyanoacrylate or CA glues, like Super/Crazy glue (as well as some other glues),  for their balsa models.
What is more important is how you apply the glue.
You said that you had a clean break which is great. Whether or not this is the case for you I would recommend not removing splinters from the break. You want to glue joint to be as free of gaps as possible. Remove splinters from the break can introduce weak points where the glue might not get the best adhesion. Make sure when you put the staircase back together that you apply glue on both sides of the break enough to cover the entire surface area.
If possible you want to clamp the work, or apply pressure in some way as clamps might be excessive for balsa, to force some of the excess glue out from the joint (you don't need a lot in the joint). Clean the extra that does come out. Let it sit for as long as the glue states on the product label.